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.
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
iii
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
iv
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
4.2 Changes in CCT (sec) with and without ES when load impedance changes from
0.64 to 0.94 (per unit) ............................................................................................ 26
4.3 Changes in CCT (sec) with and without ES when load impedance changes from
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
4.6 Load fault power changes with ES when load impedance varies from 0.64 to
v
0.94 (per unit)......................................................................................................... 30
4.7 Load fault power changes with ES when load impedance varies from 1.16 to
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
4.10 Load fault energy changes with ES when load impedance varies from 0.64 to
0.92 (per unit) …………………………………………………………………………………………………. 34
4.11 Load fault energy changes with ES when load impedance varies from 1.16 to
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
vi
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
3.8 Rotor angle versus time as impedance (per unit) varies (0.55, 0.64)..................... 16
3.9 Rotor angle versus time as impedance (per unit) varies (0.75, 0.92)..................... 17
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
vii
unit) ........................................................................................................................ 42
4.3 Rotor angle versus time with ES real (reactive) power equal to 0.05 and 0.06 (per
unit) ........................................................................................................................ 42
4.4 Rotor angle versus time with ES real (reactive) power equal to 0.07 and 0.08 (per
unit) ........................................................................................................................ 43
4.5 Rotor angle versus time with ES real (reactive) power equal to 0.09 and 0.10 (per
unit) ........................................................................................................................ 43
4.6 Rotor angle versus time with ES real (reactive) power equal to 0.2 and 0.3
(per unit)................................................................................................................. 44
viii
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.
ix
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
x
(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.
xi
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
1
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
2
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
3
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
4
Figure 1.1: Classification of Power system stability [4]
5
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
6
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)
7
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 = 𝜔𝑠𝑠 t + 𝛿𝑚
(2.4)
From equation (2.4), we obtain the angular speed of the rotor:
𝜔𝑚 = 𝜃𝑚 ′ = ω𝑠𝑠 + 𝛿𝑚 ′
8
(2.5)
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):
or
W𝑘 = 0.5 J𝜔𝑚 2 = 0.5M𝜔𝑚
(2.9)
M = (2W𝑘 /ω𝑠𝑠 )
(2.10)
9
For small changes ωm, it is reasonable to assume that M is constant, so
M= (2Wk)/ (ω𝑠𝑠 )
(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].
10
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
11
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.
12
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.
13
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].
14
Figure 3.1 SMIB without load impedance
H
5.0
𝑉1
1.0
𝑃𝑚 =𝑃1
1.0
𝜃1
17.458
𝑥1
𝑥𝑑′
0.2
0.1
𝑉2
𝜃2
0.990
11.659
𝑥2
0.2
𝑄1
𝐸′
𝛿
1.0499
0.8
28.4389
Table 3.1 initial conditions and parameters for SMIB without load impedance
15
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:
𝛿̇ =𝜔0 ∆ω=0
(3.1)
𝐸 ′ 𝑉1
2H∆ω̇ = 𝑃𝑚 sin(𝛿 − 𝜃1 ) = 0
0=0=
𝑉12
𝑥𝑑 ′
𝐸 ′ 𝑉1
𝑥𝑑′
+
𝑉1 𝐸 ′
𝑥𝑑 ′
𝑥𝑑′
cos(𝜃1 − 𝛿) -
sin(𝛿 − 𝜃1 ) -
0= -Re(𝑉2 ∠𝜃2 (
𝑉2 ∠𝜃2 ∗
))
𝑍𝐿
0 = - I𝑚 (𝑉2∠𝜃2 (
𝑉12
𝑉1 𝑉2
𝑉2 ∠𝜃2 ∗
))
𝑍𝐿
𝑥1
𝑥1
-
𝑥1
𝑉12
𝑥1
𝑉1 𝑉2
𝑥1
sin(𝜃2 − 𝜃1 )-
+
𝑉1 𝑉2
𝑥1
(3.3)
cos(𝜃1 − 𝜃2 )
sin(𝜃1 − 𝜃2 )
𝑉1 𝑉2
-
+
(3.2)
𝑉3 𝑉2
𝑥2
sin(𝜃2 − 𝜃3 )
cos(𝜃2 − 𝜃1 ) 16
(3.4)
𝑉22
𝑥2
+
𝑉3 𝑉2
𝑥2
(3.5)
cos(𝜃2 − 𝜃3 )
(3.6)
𝑃𝑚 = 𝑃𝑒 =
𝑄𝑚 = -
𝐸 ′ 𝑉1
𝑉12
𝑥𝑑 ′
𝑥𝑑′
+
(3.7)
sin(𝛿 − 𝜃1 )
𝑉1 𝐸 ′
𝑥𝑑 ′
(3.8)
cos(𝜃1 − 𝛿)
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, 𝛿̇ 𝑎𝑎𝑎 ∆ω̇ 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
5.0
𝑥1
𝑥𝑑′
𝑃𝑚 =𝑃1
1.0
0.2
0.1
𝑥2
0.2
𝑄1
0.8
Table 3.2 Initial conditions for Figure 3.2
𝐸′
δ
𝑉1
𝜃2
𝑍𝐿 (R,X)
𝜃1
6.213
0.924
0.00
0.42,0.42
7.392
𝑉2
0.932
1.23
0.48,0.48
1.1801
15.9712
1.0
1.1647
17.2792
1.0
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
17
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)(𝑍
𝑃𝑒 = 𝑃𝑚 =
𝐸 ′ 𝑉𝑡ℎ
𝑍𝐿
𝐿 +𝑋2
𝑋𝑑′ +𝑋1 +𝑍𝑡ℎ
)
sin(𝛿)
18
𝑄𝑒 = 𝑄𝑚 =
𝐸 ′ 𝑉𝑡ℎ
𝑋𝑑′ +𝑋1 +𝑍𝑡ℎ
cos(𝛿) -
𝑉𝑡ℎ 2
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.
19
II.
The fault occurs at time 𝑡𝑓 starts.
III. The line impacted by the fault is isolated by the circuit breaker at time 𝑡𝑐𝑐𝑐𝑐𝑐 .
Thus, CCT is 𝑡𝑐𝑐𝑐𝑐𝑐 - 𝑡𝑓 .
IV. The system us restored at 𝑡 = 𝑡𝑟𝑟 , and the rotor angle stabilizes at 𝑡 = 𝑡𝑝𝑝𝑝𝑝 ,
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:
20
𝛿̇ =𝜔0 ∆ω
(3.9)
𝐸 ′ 𝑉1
2H∆ω̇ = 𝑃𝑚 sin(𝛿 − 𝜃1 )-𝐾𝑑 ∆ω
0=-
0=
𝑉12
𝑥𝑑 ′
𝐸 ′ 𝑉1
𝑥𝑑′
+
𝑉1 𝐸 ′
𝑥𝑑′
𝑥𝑑 ′
sin(𝛿 − 𝜃1 ) 𝑉 ∠𝜃2 ∗
))
1 ∥𝑍𝐿
0= -Re(𝑉2 ∠𝜃2 (𝑗𝑥2
𝐸 ′ 𝑉1
𝑥𝑑′
2𝑥1
𝑉1 𝑉2
2𝑥1
𝑉 ∠𝜃2 ∗
))
1 ∥𝑍𝐿
0 = - I𝑚 (𝑉2∠𝜃2 (𝑗𝑥2
𝑃𝑚 = 1.0≠𝑃𝑒 =
𝑉12
cos(𝜃1 − 𝛿) -
-
𝑉1 𝑉2
2𝑥1
cos(𝜃1 − 𝜃2 )- I𝑚 (𝑉1 ∠𝜃1 (
2𝑥1
𝑉12
2𝑥1
sin(𝜃2 − 𝜃1 )-
+
𝑉1 𝑉2
2𝑥1
𝑉3 𝑉2
𝑥2
𝑉1 ∠𝜃1 ∗
))
𝑗𝑥1
(3.11)
𝑉1 ∠𝜃1 ∗
))
𝑗𝑥1
(3.12)
sin(𝜃2 − 𝜃3 )
(3.13)
sin(𝜃1 − 𝜃2 ) - 𝑅𝑒 (𝑉1 ∠𝜃1 (
𝑉1 𝑉2
-
+
(3.10)
cos(𝜃2 − 𝜃1 ) -
𝑉22
𝑥2
+
𝑉3 𝑉2
𝑥2
cos(𝜃2 − 𝜃3 )
(3.14)
(3.15)
sin(𝛿 − 𝜃1 )
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.
21
Figure 3.5 Simplified diagram during faulted operation
Figure 3.6 Post-fault circuit
Postfault DAE:
𝛿̇ =𝜔0 ∆ω
(3.16)
22
𝐸 ′ 𝑉1
2H∆ω̇ = 𝑃𝑚 sin(𝛿 − 𝜃1 )
0=0=
𝑉12
𝑥𝑑 ′
𝐸 ′ 𝑉1
𝑥𝑑 ′
+
𝑉1 𝐸 ′
𝑥𝑑 ′
𝑥𝑑′
𝑉12
cos(𝜃1 − 𝛿) -
sin(𝛿 − 𝜃1 ) -
0= -Re(𝑉2 ∠𝜃2 (
𝑉2 ∠𝜃2 ∗
))
𝑍𝐿
0 = - I𝑚 (𝑉2∠𝜃2 (
𝑉1 𝑉2
𝑉2 ∠𝜃2 ∗
))
𝑍𝐿
(3.17)
2𝑥1
2𝑥1
-
𝑉1 𝑉2
2𝑥1
2𝑥1
𝑉12
2𝑥1
sin(𝜃2 − 𝜃1 )-
+
𝑉1 𝑉2
2𝑥1
(3.18)
cos(𝜃1 − 𝜃2 )
sin(𝜃1 − 𝜃2 )
𝑉1 𝑉2
-
+
𝑉3 𝑉2
𝑥2
(3.19)
sin(𝜃2 − 𝜃3 )
cos(𝜃2 − 𝜃1 ) -
𝑉22
𝑥2
+
𝑉3 𝑉2
𝑥2
(3.20)
cos(𝜃2 − 𝜃3 )
(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.
23
Figure 3.7 Impedance (p.u.) = (0.42, 0.48), rotor angle v.s. time, CCT, stable
post-fault system.
24
Figure 3.8 Impedance (p.u.) = (0.55, 0.64), rotor angle v.s. time, CCT, stable
post-fault system.
25
Figure 3.9 Impedance (p.u.) = (0.75, 0.92), rotor angle v.s. time, CCT, stable
post-fault system
26
Figure 3.10 Impedance (p.u.) = (1.16, 1.57), rotor angle v.s. time, CCT, stable
post-fault system.
27
Figure 3.11 Impedance (p.u.) = (2.39, 4.84), rotor angle v.s. time, CCT, stable state
after fault.
28
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
CCT (sec) 0.2518 0.2508 0.2498 0.2487 0.2478 0.2468 0.2458 0.2448 0.2439
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.
29
4.84
0.2429
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:
30
0=0=
𝑉12
𝑥𝑑 ′
𝐸 ′ 𝑉1
𝑥𝑑′
+
𝑉1 𝑉2
0= 0=-
𝑥1
𝑉12
𝑥1
𝑥𝑑 ′
𝑉1 𝑉2
𝑉1 𝑉2
𝑉3 𝑉2
𝑉12
𝑥𝑑 ′
𝑥𝑑′
+
+
𝑉1 𝑉2
𝑥1
(3.22)
cos(𝜃1 − 𝜃2 )
(3.23)
sin(𝜃1 − 𝜃2 )
sin(𝜃2 − 𝜃3 )
𝑥2
cos(𝜃2 − 𝜃1 ) -
𝑥1
𝐸 ′ 𝑉1
𝑥1
𝑥1
sin(𝜃2 − 𝜃1 )-
+
𝑉12
cos(𝜃1 − 𝛿) -
sin(𝛿 − 𝜃1 ) -
𝑃𝑚 = 𝑃𝑒 =
𝑄𝑚 = -
𝑉1 𝐸 ′
𝑉22
𝑥2
+
𝑉3 𝑉2
𝑥2
(3.24)
cos(𝜃2 − 𝜃3 )
(3.25)
(3.26)
sin(𝛿 − 𝜃1 )
𝑉1 𝐸 ′
𝑥𝑑 ′
(3.27)
cos(𝜃1 − 𝛿)
During the fault, the DAE is as follows:
0=-
0=
𝑉12
𝑥𝑑 ′
𝐸 ′ 𝑉1
𝑥𝑑′
+
𝑉1 𝐸 ′
𝑥𝑑 ′
𝑉12
cos(𝜃1 − 𝛿) -
sin(𝛿 − 𝜃1 ) -
0= -Re(𝑉2 ∠𝜃2 (
𝑉2 ∠𝜃2 ∗
))
𝑗𝑥1
0 = - I𝑚 (𝑉2∠𝜃2 (
2𝑥1
𝑉1 𝑉2
𝑉2 ∠𝜃2 ∗
))
𝑗𝑥1
2𝑥1
-
𝑉1 𝑉2
2𝑥1
cos(𝜃1 − 𝜃2 )- I𝑚 (𝑉1 ∠𝜃1 (
sin(𝜃1 − 𝜃2 ) - 𝑅𝑒 (𝑉1 ∠𝜃1 (
𝑉1 𝑉2
-
+
2𝑥1
𝑉12
2𝑥1
sin(𝜃2 − 𝜃1 )-
+
𝑉1 𝑉2
2𝑥1
𝑉3 𝑉2
𝑥2
𝑉1 ∠𝜃1 ∗
))
𝑗𝑥1
𝑉1 ∠𝜃1 ∗
))
𝑗𝑥1
sin(𝜃2 − 𝜃3 )
cos(𝜃2 − 𝜃1 ) -
𝑉22
𝑥2
+
𝑉3 𝑉2
𝑥2
(3.28)
(3.29)
(3.30)
cos(𝜃2 − 𝜃3 )
(3.31)
During post-fault, the DAE is as follows:
𝛿̇ =𝜔0 ∆ω
(3.32)
𝐸 ′ 𝑉1
2H∆ω̇ = 𝑃𝑚 sin(𝛿 − 𝜃1 )
𝑥𝑑′
(3.33)
31
0=0=
𝑉12
𝑥𝑑 ′
𝐸 ′ 𝑉1
0= 0=-
+
𝑥𝑑′
𝑉1 𝑉2
2𝑥1
𝑉12
2𝑥1
𝑉1 𝐸 ′
𝑥𝑑 ′
cos(𝜃1 − 𝛿) -
sin(𝛿 − 𝜃1 ) -
sin(𝜃2 − 𝜃1 )-
+
𝑉1 𝑉2
2𝑥1
𝑉1 𝑉2
2𝑥1
𝑉12
2𝑥1
𝑉3 𝑉2
𝑥2
+
𝑉1 𝑉2
2𝑥1
cos(𝜃1 − 𝜃2 )
sin(𝜃1 − 𝜃2 )
sin(𝜃2 − 𝜃3 )
cos(𝜃2 − 𝜃1 ) -
𝑉22
𝑥2
+
𝑉3 𝑉2
𝑥2
(3.35)
(3.36)
cos(𝜃2 − 𝜃3 )
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
32
(3.34)
(3.37)
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.
33
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
34
(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
35
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:
𝛿̇ =𝜔0 ∆ω
(4.1)
𝐸 ′ 𝑉1
2H∆ω̇ = 𝑃𝑚 sin(𝛿 − 𝜃1 )
0=0=
𝑉12
𝑥𝑑 ′
𝐸 ′ 𝑉1
𝑥𝑑′
+
𝑉1 𝐸 ′
𝑥𝑑 ′
𝑥𝑑′
𝑉12
cos(𝜃1 − 𝛿) -
sin(𝛿 − 𝜃1 ) -
0= -Re(𝑉2 ∠𝜃2 (
𝑉2 ∠𝜃2 ∗
))
𝑍𝐿
0 = - I𝑚 (𝑉2∠𝜃2 (
𝑉1 𝑉2
𝑉2 ∠𝜃2 ∗
))
𝑍𝐿
(4.2)
2𝑥1
2𝑥1
-
𝑉1 𝑉2
2𝑥1
2𝑥1
𝑉12
2𝑥1
sin(𝜃2 − 𝜃1 )-
+
𝑉1 𝑉2
2𝑥1
(4.3)
cos(𝜃1 − 𝜃2 )
sin(𝜃1 − 𝜃2 )
𝑉1 𝑉2
-
+
𝑉3 𝑉2
𝑥2
(4.4)
sin(𝜃2 − 𝜃3 )
cos(𝜃2 − 𝜃1 ) -
𝑉22
𝑥2
+
(4.5)
𝑉3 𝑉2
𝑥2
cos(𝜃2 − 𝜃3 )
(4.6)
Fault DAE:
𝛿̇ =𝜔0 ∆ω
(4.7)
𝐸 ′ 𝑉1
2H∆ω̇ = 𝑃𝑚 sin(𝛿 − 𝜃1 )-𝐾𝑑 ∆ω
𝑉2
𝑉 𝐸′
𝑥𝑑′
𝑉2
(4.8)
𝑉1 𝑉2
0 = - 𝑥1 + 𝑥1 cos(𝜃1 − 𝛿) - 1 +
cos(𝜃1 − 𝜃2 )- I𝑚 (𝑉1 ∠𝜃1 (
2𝑥1
2𝑥1
′
′
𝑑
𝑑
36
𝑉1 ∠𝜃1 ∗
) )-𝑄𝐸𝐸
𝑗𝑥1
(4.9)
0=
𝐸 ′ 𝑉1
𝑥𝑑′
sin(𝛿 − 𝜃1 ) -
𝑉1 𝑉2
2𝑥1
𝑉 ∠𝜃2 ∗
))
1 ∥𝑍𝐿
0= -Re(𝑉2 ∠𝜃2 (𝑗𝑥2
𝑉 ∠𝜃2 ∗
))
1 ∥𝑍𝐿
𝐸 ′ 𝑉1
𝑥𝑑′
𝑉1 𝑉2
-
0 = - I𝑚 (𝑉2∠𝜃2 (𝑗𝑥2
𝑃𝑚 = 1.0≠𝑃𝑒 =
sin(𝜃1 − 𝜃2 ) - 𝑅𝑒 (𝑉1 ∠𝜃1 (
-
2𝑥1
𝑉12
2𝑥1
sin(𝜃2 − 𝜃1 )-
+
𝑉1 𝑉2
2𝑥1
𝑉3 𝑉2
𝑥2
𝑉1 ∠𝜃1 ∗
))
𝑗𝑥1
-𝑃𝐸𝐸
sin(𝜃2 − 𝜃3 )
cos(𝜃2 − 𝜃1 ) -
𝑉22
𝑥2
+
𝑉3 𝑉2
𝑥2
(4.10)
(4.11)
cos(𝜃2 − 𝜃3 )
(4.12)
(4.13)
sin(𝛿 − 𝜃1 )
Post-fault DAE:
𝛿̇ =𝜔0 ∆ω
(4.14)
𝐸 ′ 𝑉1
2H∆ω̇ = 𝑃𝑚 sin(𝛿 − 𝜃1 )
0=0=
𝑉12
𝑥𝑑 ′
𝐸 ′ 𝑉1
𝑥𝑑′
+
𝑉1 𝐸 ′
𝑥𝑑 ′
𝑥𝑑′
𝑉12
cos(𝜃1 − 𝛿) -
sin(𝛿 − 𝜃1 ) -
0= -Re(𝑉2 ∠𝜃2 (
𝑉2 ∠𝜃2 ∗
))
𝑍𝐿
0 = - I𝑚 (𝑉2∠𝜃2 (
𝑉1 𝑉2
𝑉2 ∠𝜃2 ∗
))
𝑍𝐿
(4.15)
2𝑥1
2𝑥1
-
𝑉1 𝑉2
2𝑥1
2𝑥1
𝑉12
2𝑥1
sin(𝜃2 − 𝜃1 )-
+
𝑉1 𝑉2
2𝑥1
(4.16)
cos(𝜃1 − 𝜃2 )
sin(𝜃1 − 𝜃2 )
𝑉1 𝑉2
-
+
𝑉3 𝑉2
𝑥2
(4.17)
sin(𝜃2 − 𝜃3 )
cos(𝜃2 − 𝜃1 ) -
𝑉22
𝑥2
+
𝑉3 𝑉2
𝑥2
(4.18)
cos(𝜃2 − 𝜃3 )
(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.
37
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.
38
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.).
39
ES
POWER
LOAD
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.).
40
ES
POWER
LOAD
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.).
41
ES
POWER
LOAD
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
42
how the power to the load changes during the fault with different amounts of ES
power.
ES
POWER
LOAD
0.42
0.48
0.55
0.00
0.0299 (P=Q)
(fault power)
0.0261(P=Q)
(fault power)
0.0266(P=Q)
(fault power)
0.03
0.0307(P=Q)
(fault power)
0.0267(P=Q)
(fault power)
0.0231(P=Q)
(fault power)
0.04
0.0310(P=Q)
(fault power)
0.0269(P=Q)
(fault power)
0.0233(P=Q)
(fault power)
0.05
0.0311(P=Q)
(fault power)
0.0271(P=Q)
(fault power)
0.0234(P=Q)
(fault power)
0.06
0.0312(P=Q)
(fault power)
0.0274(P=Q)
(fault power)
0.0236(P=Q)
(fault power)
0.07
0.0314(P=Q)
(fault power)
0.0276(P=Q)
(fault power)
0.0238(P=Q)
(fault power)
0.08
0.0316(P=Q)
(fault power)
0.0278(P=Q)
(fault power)
0.0240(P=Q)
(fault power)
0.09
0.0317(P=Q)
(fault power)
0.0280(P=Q)
(fault power)
0.0242(P=Q)
(fault power)
0.10
0.0319(P=Q)
(fault power)
0.0283(P=Q)
(fault power)
0.0244(P=Q)
(fault power)
0.20
0.0339(P=Q)
(fault power)
0.0301(P=Q)
(fault power)
0.0265(P=Q)
(fault power)
0.30
0.0360(P=Q)
(fault power)
0.0318(P=Q)
(fault power)
0.0279(P=Q)
(fault power)
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.).
43
ES
POWER
LOAD
0.64
0.75
0.92
0.00
0.0195 (P=Q)
(fault power)
0.0165(P=Q)
(fault power)
0.0135(P=Q)
(fault power)
0.03
0.0199(P=Q)
(fault power)
0.0168(P=Q)
(fault power)
0.0138(P=Q)
(fault power)
0.04
0.0200(P=Q)
(fault power)
0.0169(P=Q)
(fault power)
0.0138(P=Q)
(fault power)
0.05
0.0201(P=Q)
(fault power)
0.0170(P=Q)
(fault power)
0.0139(P=Q)
(fault power)
0.06
0.0202(P=Q)
(fault power)
0.0171(P=Q)
(fault power)
0.0140(P=Q)
(fault power)
0.07
0.0204(P=Q)
(fault power)
0.0172(P=Q)
(fault power)
0.0141(P=Q)
(fault power)
0.08
0.0205(P=Q)
(fault power)
0.0173(P=Q)
(fault power)
0.0142(P=Q)
(fault power)
0.09
0.0207(P=Q)
(fault power)
0.0174(P=Q)
(fault power)
0.0143(P=Q)
(fault power)
0.10
0.0208(P=Q)
(fault power)
0.0175(P=Q)
(fault power)
0.0114(P=Q)
(fault power)
0.20
0.0224(P=Q)
(fault power)
0.0187(P=Q)
(fault power)
0.0153(P=Q)
(fault power)
0.30
0.0240(P=Q)
(fault power)
0.0202(P=Q)
(fault power)
0.0163(P=Q)
(fault power)
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.).
44
ES
POWER
LOAD
L
1.16
1.57
2.39
0.00
0.0106 (P=Q)
(fault power)
0.0079(P=Q)
(fault power)
0.0052(P=Q)
(fault power)
0.03
0.0108(P=Q)
(fault power)
0.0080(P=Q)
(fault power)
0.0053(P=Q)
(fault power)
0.04
0.0109(P=Q)
(fault power)
0.0081(P=Q)
(fault power)
0.0053(P=Q)
(fault power)
0.05
0.0110(P=Q)
(fault power)
0.0081(P=Q)
(fault power)
0.0054(P=Q)
(fault power)
0.06
0.0111(P=Q)
(fault power)
0.0082(P=Q)
(fault power)
0.0054(P=Q)
(fault power)
0.07
0.0111(P=Q)
(fault power)
0.0082(P=Q)
(fault power)
0.0054(P=Q)
(fault power)
0.08
0.0112(P=Q)
(fault power)
0.0083(P=Q)
(fault power)
0.0054(P=Q)
(fault power)
0.09
0.0113(P=Q)
(fault power)
0.0084(P=Q)
(fault power)
0.0054(P=Q)
(fault power)
0.10
0.0113(P=Q)
(fault power)
0.0084(P=Q)
(fault power)
0.0055(P=Q)
(fault power)
0.20
0.0121(P=Q)
(fault power)
0.0089(P=Q)
(fault power)
0.0059(P=Q)
(fault power)
0.30
0.1280(P=Q)
(fault power)
0.0094(P=Q)
(fault power)
0.0062(P=Q)
(fault power)
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.).
45
ES
POWER
LOAD
4.84
0.00
0.0026(P=Q)
(fault power)
0.03
0.0026(P=Q)
(fault power)
0.04
0.0026(P=Q)
(fault power)
0.05
0.0026(P=Q)
(fault power)
0.06
0.0027(P=Q)
(fault power)
0.07
0.0027(P=Q)
(fault power)
0.08
0.0027(P=Q)
(fault power)
0.09
0.0027(P=Q)
(fault power)
0.10
0.0027(P=Q)
(fault power)
0.20
0.0029(P=Q)
(fault power)
0.30
0.0031(P=Q)
(fault power)
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
46
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
POWER
LOAD
0.42
0.48
0.55
0.00
7.5 KJ
6.5 KJ
(fault energy) (fault energy)
5.6 KJ
(fault energy)
0.03
7.9 KJ
6.9 KJ
(fault energy) (fault energy)
5.9 KJ
(fault energy)
0.04
8.0 KJ
7.0 KJ
(fault energy) (fault energy)
6.0 KJ
(fault energy)
0.05
8.1 KJ
7.1 KJ
(fault energy) (fault energy)
6.1 KJ
(fault energy)
0.06
8.2 KJ
7.2 KJ
(fault energy) (fault energy)
6.2 KJ
(fault energy)
0.07
8.3 KJ
7.3 KJ
(fault energy) (fault power)
6.3 KJ
(fault energy)
0.08
8.5 KJ
7.4 KJ
(fault energy) (fault energy)
6.4 KJ
(fault energy)
0.09
8.6 KJ
7.5 KJ
(fault energy) (fault energy)
6.5 KJ
(fault energy)
0.10
8.7 KJ
7.7 KJ
(fault energy) (fault energy)
6.6 KJ
(fault energy)
0.20
10.0 KJ
8.8 KJ
(fault energy) (fault energy)
7.8 KJ
(fault energy)
0.30
11.6 KJ
10.2 KJ
(fault energy) (fault energy)
8.9 KJ
(fault energy)
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.).
47
ES
POWER
LOAD
0.64
0.75
0.92
0.00
4.8 KJ
4.1 KJ
(fault energy) (fault energy)
3.3 KJ
(fault energy)
0.03
5.1 KJ
4.3 KJ
(fault energy) (fault energy)
3.5 KJ
(fault energy)
0.04
5.1 KJ
4.3 KJ
(fault energy) (fault energy)
3.5 KJ
(fault energy)
0.05
5.2 KJ
4.3 KJ
(fault energy) (fault energy)
3.6 KJ
(fault energy)
0.06
5.3 KJ
4.4 KJ
(fault energy) (fault energy)
3.6 KJ
(fault energy)
0.07
5.4 KJ
4.5 KJ
(fault energy) (fault power)
3.7 KJ
(fault energy)
0.08
5.4 KJ
4.6 KJ
(fault energy) (fault energy)
3.7 KJ
(fault energy)
0.09
5.5 KJ
4.6 KJ
(fault energy) (fault energy)
3.8 KJ
(fault energy)
0.10
5.6 KJ
4.7 KJ
(fault energy) (fault energy)
3.8 KJ
(fault energy)
0.20
6.5 KJ
5.4 KJ
(fault energy) (fault energy)
4.4 KJ
(fault energy)
0.30
7.6 KJ
6.4 KJ
(fault energy) (fault energy)
5.1 KJ
(fault energy)
Table 4.10 Energy delivered to the load during the fault with ES power when load
impedance is from 0.64 to 0.92 (unit of ES POWER and LOAD: p.u.).
48
ES
POWER
LOAD
1.16
1.57
2.39
0.00
2.6 KJ
1.9 KJ
(fault energy) (fault energy)
1.3 KJ
(fault energy)
0.03
2.7 KJ
2.0 KJ
(fault energy) (fault energy)
1.3 KJ
(fault energy)
0.04
2.8 KJ
2.0 KJ
(fault energy) (fault energy)
1.3 KJ
(fault energy)
0.05
2.8 KJ
2.1 KJ
(fault energy) (fault energy)
1.4 KJ
(fault energy)
0.06
2.9 KJ
2.1 KJ
(fault energy) (fault energy)
1.4 KJ
(fault energy)
0.07
2.9 KJ
2.1 KJ
(fault energy) (fault power)
1.4 KJ
(fault energy)
0.08
2.9 KJ
2.2 KJ
(fault energy) (fault energy)
1.4 KJ
(fault energy)
0.09
3.0 KJ
2.2 KJ
(fault energy) (fault energy)
1.4 KJ
(fault energy)
0.10
3.0 KJ
2.2 KJ
(fault energy) (fault energy)
1.4 KJ
(fault energy)
0.20
3.5 KJ
2.5 KJ
(fault energy) (fault energy)
1.7 KJ
(fault energy)
0.30
4.0 KJ
2.9 KJ
(fault energy) (fault energy)
1.9 KJ
(fault energy)
Table 4.11 Energy delivered 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.).
49
ES
POWER
LOAD
4.84
0.00
0.6 KJ
(fault energy)
0.03
0.6 KJ
(fault energy)
0.04
0.7 KJ
(fault energy)
0.05
0.7 KJ
(fault energy)
0.06
0.7 KJ
(fault energy)
0.07
0.7 KJ
(fault energy)
0.08
0.7 KJ
(fault energy)
0.09
0.7 KJ
(fault energy)
0.10
0.7 KJ
(fault energy)
0.20
0.8 KJ
(fault energy)
0.30
0.9 KJ
(fault energy)
Table 4.12 Energy delivered to the load during the fault with ES power when load
impedance is 4.84 (unit of ES POWER and LOAD: p.u.).
50
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.
51
ES
POWER
LOAD
0.42
0.48
0.55
0.00
0.0 KJ
0.0 KJ
(fault energy) (fault energy)
0.0 KJ
(fault energy)
0.03
7.7 KJ
7.7 KJ
(fault energy) (fault energy)
7.7 KJ
(fault energy)
0.04
10.4 KJ
10.3 KJ
(fault energy) (fault energy)
10.3 KJ
(fault energy)
0.05
13.1 KJ
13.0 KJ
(fault energy) (fault energy)
13.0 KJ
(fault energy)
0.06
15.8 KJ
15.8 KJ
(fault energy) (fault energy)
15.7 KJ
(fault energy)
0.07
18.6 KJ
18.5 KJ
(fault energy) (fault power)
18.4 KJ
(fault energy)
0.08
21.4 KJ
21.3 KJ
(fault energy) (fault energy)
21.2 KJ
(fault energy)
0.09
24.3 KJ
24.2 KJ
(fault energy) (fault energy)
24.1 KJ
(fault energy)
0.10
27.2 KJ
27.1 KJ
(fault energy) (fault energy)
27.0 KJ
(fault energy)
0.20
59.1 KJ
58.8 KJ
(fault energy) (fault energy)
58.5 KJ
(fault energy)
0.30
96.7 KJ
96.1 KJ
(fault energy) (fault energy)
95.5 KJ
(fault energy)
Table 4.13 Energy absorbed by the ES power system during the fault when load
impedance is from 0.42 to 0.55 (unit of ES POWER and LOAD: p.u.).
52
ES
POWER
LOAD
0.64
0.75
0.92
0.00
0.0 KJ
0.0 KJ
(fault energy) (fault energy)
0.0 KJ
(fault energy)
0.03
7.6 KJ
7.6 KJ
(fault energy) (fault energy)
7.6 KJ
(fault energy)
0.04
10.3 KJ
10.2 KJ
(fault energy) (fault energy)
10.2 KJ
(fault energy)
0.05
12.9 KJ
12.9 KJ
(fault energy) (fault energy)
12.8 KJ
(fault energy)
0.06
15.6 KJ
15.6 KJ
(fault energy) (fault energy)
15.5 KJ
(fault energy)
0.07
18.4 KJ
18.3 KJ
(fault energy) (fault power)
18.2 KJ
(fault energy)
0.08
21.2 KJ
21.1 KJ
(fault energy) (fault energy)
21.0 KJ
(fault energy)
0.09
24.0 KJ
23.9 KJ
(fault energy) (fault energy)
23.8 KJ
(fault energy)
0.10
26.9 KJ
26.8 KJ
(fault energy) (fault energy)
26.6 KJ
(fault energy)
0.20
58.2 KJ
58.0 KJ
(fault energy) (fault energy)
57.7 KJ
(fault energy)
0.30
94.9 KJ
94.4 KJ
(fault energy) (fault energy)
93.8 KJ
(fault energy)
Table 4.14 Energy absorbed by the ES power system during the fault when load
impedance is from 0.64 to 0.92 (unit of ES POWER and LOAD: p.u.).
53
.
ES
POWER
LOAD
1.16
1.57
2.39
0.00
0.00 KJ
0.00 KJ
(fault energy) (fault energy)
0.00 KJ
(fault energy)
0.03
7.6 KJ
7.5 KJ
(fault energy) (fault energy)
7.5 KJ
(fault energy)
0.04
10.1 KJ
10.1 KJ
(fault energy) (fault energy)
10.1 KJ
(fault energy)
0.05
12.8 KJ
12.7 KJ
(fault energy) (fault energy)
12.7 KJ
(fault energy)
0.06
15.4 KJ
15.4 KJ
(fault energy) (fault energy)
15.3 KJ
(fault energy)
0.07
18.1 KJ
18.1 KJ
(fault energy) (fault power)
18.0 KJ
(fault energy)
0.08
20.9 KJ
20.8 KJ
(fault energy) (fault energy)
20.7 KJ
(fault energy)
0.09
23.7 KJ
23.6 KJ
(fault energy) (fault energy)
23.5 KJ
(fault energy)
0.10
26.5 KJ
26.4 KJ
(fault energy) (fault energy)
26.3 KJ
(fault energy)
0.20
57.4 KJ
57.1 KJ
(fault energy) (fault energy)
56.9 KJ
(fault energy)
0.30
93.3 KJ
92.8 KJ
(fault energy) (fault energy)
92.2 KJ
(fault energy)
Table 4.14 Energy absorbed by the ES power system during the fault when load
impedance is from 1.16 to 2.39 (unit of ES POWER and LOAD: p.u.).
54
ES
POWER
LOAD
4.84
0.00
0.00 KJ
(fault energy)
0.03
7.50 KJ
(fault energy)
0.04
10.0 KJ
(fault energy)
0.05
12.6 KJ
(fault energy)
0.06
15.2 KJ
(fault energy)
0.07
17.9 KJ
(fault energy)
0.08
20.6 KJ
(fault energy)
0.09
23.4 KJ
(fault energy)
0.10
26.2 KJ
(fault energy)
0.20
56.6 KJ
(fault energy)
0.30
91.7 KJ
(fault energy)
Table 4.15 Energy absorbed by the ES power system during the fault when load
impedance is 4.84 (unit of ES POWER and LOAD: p.u.).
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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.).
56
Figure 4.3 Rotor angle versus time with ES power equal to 0.05 and 0.06 (p.u.).
Figure 4.4 Rotor angle versus time with ES power equal to 0.07 and 0.08 (p.u.).
57
Figure 4.5 Rotor angle versus time with ES power equal to 0.09 and 0.1 (p.u.).
58
Figure 4.6 Rotor angle versus time with ES power equal to 0.2 and 0.3 (p.u.).
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
59
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
60
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
61
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