TRANSIENT STABILITY ANALYSIS OF POWER SYSTEMS WITH ENERGY STORAGE

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 ω sm, electromagnetic torque T e, and mechanical torque T m means mechanical torque. In steady-state, T m =T e (2.1) 7

When a disturbance occurs, the torque deviates from steady-state, causing an accelerating (T m >T e ) or decelerating (T m <T e ) torque: T a (accelerating torque) = T m - T e (2.2) Assume J is the combined inertia of generator and prime mover, neglecting friction and damping torque we have: J θ m = T a = T m - T e (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: θ m = ω ss t + δ m (2.4) From equation (2.4), we obtain the angular speed of the rotor: ω m = θ m = ω ss + δ m 8 (2.5)

Where θ m = δ m (2.6) Substituting (2.6) into (2.3), we obtain: J δ m = T a = T m - T e (2.7) Multiply eq. (2.7) by ω m : ω m J δ m = ω m T m - ω m T e = P m - P e (2.8) J ω m is called the constant of inertia, referenced by M and associated with W k (kinetic energy): W k = 0.5 Jω m 2 = 0.5Mω m (2.9) or M = (2W k /ω ss ) (2.10) 9

For small changes ω m, it is reasonable to assume that M is constant, so M= (2Wk)/ (ω ss ) (2.11) Then we obtain the standard from of the swing equation: M δ m = P m - P e (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 P m =P 1 x d x 1 x 2 Q 1 5.0 1.0 0.2 0.1 0.2 0.8 V 1 θ 1 V 2 θ 2 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 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: δ =ω 0 ω= 0 (3.1) 2H ω = P m - E V 1 x d 0 = - V 1 2 x d sin(δ θ 1 ) = 0 (3.2) + V 1E cos(θ x 1 δ) - V 1 2 + V 1V 2 cos(θ d x 1 x 1 θ 2 ) (3.3) 1 0 = E V 1 x d sin(δ θ 1 ) - V 1V 2 x 1 sin(θ 1 θ 2 ) (3.4) 0= -Re(V 2 θ 2 ( V 2 θ 2 Z L ) ) - V 1V 2 x 1 sin(θ 2 θ 1 )- V 3V 2 x 2 sin(θ 2 θ 3 ) (3.5) 0 = - I m (V 2 θ 2 ( V 2 θ 2 ) ) - V 2 1 Z L x 1 + V 1V 2 cos(θ x 2 θ 1 ) - V 2 2 1 16 x 2 + V 3V 2 x 2 cos(θ 2 θ 3 ) (3.6)

P m = P e = E V 1 x d sin(δ θ 1 ) (3.7) Q m = - V 1 2 x d + V 1E cos(θ x 1 δ) (3.8) d Bus 2 is the constant impedance load, bus 3 is the slack bus V 3 =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, δ aaa ω are zero. 2H (p.u.) is the constant mechanical inertia of the generator, and Z L (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 P m =P 1 x d x 1 x 2 Q 1 5.0 1.0 0.2 0.1 0.2 0.8 Table 3.2 Initial conditions for Figure 3.2 E δ V 1 θ 1 V 2 θ 2 Z L (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 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, V 1, V 2, Z L : p. u, unit of δ, θ 1, θ 2 : degree) An alternate approach is to determine E, δ and then substitute these values into equations (3.5) to (3.8) to obtain (V 1, V 2, θ 1, θ 2 ). 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 Z th = Z L X 2 V th = (1 0)( Z L ) Z L +X 2 P e = P m = E V th X d +X 1 +Z th sin(δ) 18

Q e = Q m = E V th X d +X 1 +Z th cos(δ) - V th 2 X d +X 1 +Z th 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 x 1 is replaced by one parallel branch of line 2x 1, 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 t f starts. III. The line impacted by the fault is isolated by the circuit breaker at time t ccccc. Thus, CCT is t ccccc - t f. IV. The system us restored at t = t rr, and the rotor angle stabilizes at t = t pppp, 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 2x 1 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

δ =ω 0 ω (3.9) 2H ω = P m - E V 1 x d 0 = - V 1 2 x d sin(δ θ 1 )-K d ω (3.10) + V 1E cos(θ x 1 δ) - V 1 2 + V 1V 2 cos(θ d 2x 1 2x 1 θ 2 )- I m (V 1 θ 1 ( V 1 θ 1 ) ) 1 jx 1 (3.11) 0 = E V 1 x d sin(δ θ 1 ) - V 1V 2 2x 1 sin(θ 1 θ 2 ) - R e (V 1 θ 1 ( V 1 θ 1 jx 1 ) ) (3.12) 0= -Re(V 2 θ 2 ( V 2 θ 2 jx 1 Z L ) ) - V 1V 2 2x 1 sin(θ 2 θ 1 )- V 3V 2 x 2 sin(θ 2 θ 3 ) (3.13) 0 = - I m (V 2 θ 2 ( V 2 θ 2 ) ) - V 2 1 jx 1 Z L 2x 1 + V 1V 2 cos(θ 2x 2 θ 1 ) - V 2 2 1 x 2 + V 3V 2 x 2 cos(θ 2 θ 3 ) (3.14) P m = 1.0 P e = E V 1 x d sin(δ θ 1 ) (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 (V 1, V 2, θ 1, θ 2 ), and the input variables are (E, P m ). 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: δ =ω 0 ω (3.16) 22

2H ω = P m - E V 1 x d 0 = - V 1 2 x d sin(δ θ 1 ) (3.17) + V 1E cos(θ x 1 δ) - V 1 2 + V 1V 2 cos(θ d 2x 1 2x 1 θ 2 ) (3.18) 1 0 = E V 1 x d sin(δ θ 1 ) - V 1V 2 2x 1 sin(θ 1 θ 2 ) (3.19) 0= -Re(V 2 θ 2 ( V 2 θ 2 Z L ) ) - V 1V 2 2x 1 sin(θ 2 θ 1 )- V 3V 2 x 2 sin(θ 2 θ 3 ) (3.20) 0 = - I m (V 2 θ 2 ( V 2 θ 2 ) ) - V 2 1 Z L 2x 1 + V 1V 2 cos(θ 2x 2 θ 1 ) - V 2 2 1 x 2 + V 3V 2 x 2 cos(θ 2 θ 3 ) (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 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. 29

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 = - V 1 2 x d + V 1E cos(θ x 1 δ) - V 1 2 + V 1V 2 cos(θ d x 1 x 1 θ 2 ) (3.22) 1 0 = E V 1 x d sin(δ θ 1 ) - V 1V 2 x 1 sin(θ 1 θ 2 ) (3.23) 0= - V 1V 2 x 1 sin(θ 2 θ 1 )- V 3V 2 x 2 sin(θ 2 θ 3 ) (3.24) 0 = - V 1 2 x 1 + V 1V 2 cos(θ x 2 θ 1 ) - V 2 2 1 x 2 + V 3V 2 x 2 cos(θ 2 θ 3 ) (3.25) P m = P e = E V 1 x d sin(δ θ 1 ) (3.26) Q m = - V 1 2 x d + V 1E cos(θ x 1 δ) (3.27) d During the fault, the DAE is as follows: 0 = - V 1 2 x d + V 1E cos(θ x 1 δ) - V 1 2 + V 1V 2 cos(θ d 2x 1 2x 1 θ 2 )- I m (V 1 θ 1 ( V 1 θ 1 ) ) 1 jx 1 (3.28) 0 = E V 1 x d sin(δ θ 1 ) - V 1V 2 2x 1 sin(θ 1 θ 2 ) - R e (V 1 θ 1 ( V 1 θ 1 jx 1 ) ) (3.29) 0= -Re(V 2 θ 2 ( V 2 θ 2 jx 1 ) ) - V 1V 2 2x 1 sin(θ 2 θ 1 )- V 3V 2 x 2 sin(θ 2 θ 3 ) (3.30) 0 = - I m (V 2 θ 2 ( V 2 θ 2 ) ) - V 2 1 jx 1 2x 1 + V 1V 2 cos(θ 2x 2 θ 1 ) - V 2 2 1 x 2 + V 3V 2 x 2 cos(θ 2 θ 3 ) (3.31) During post-fault, the DAE is as follows: δ =ω 0 ω (3.32) 2H ω = P m - E V 1 x d sin(δ θ 1 ) (3.33) 31

0 = - V 1 2 x d + V 1E cos(θ x 1 δ) - V 1 2 + V 1V 2 cos(θ d 2x 1 2x 1 θ 2 ) (3.34) 1 0 = E V 1 x d sin(δ θ 1 ) - V 1V 2 2x 1 sin(θ 1 θ 2 ) (3.35) 0= - V 1V 2 2x 1 sin(θ 2 θ 1 )- V 3V 2 x 2 sin(θ 2 θ 3 ) (3.36) 0 = - V 1 2 2x 1 + V 1V 2 cos(θ 2x 2 θ 1 ) - V 2 2 1 x 2 + V 3V 2 x 2 cos(θ 2 θ 3 ) (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 32

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: δ =ω 0 ω (4.1) 2H ω = P m - E V 1 x d 0 = - V 1 2 x d sin(δ θ 1 ) (4.2) + V 1E cos(θ x 1 δ) - V 1 2 + V 1V 2 cos(θ d 2x 1 2x 1 θ 2 ) (4.3) 1 0 = E V 1 x d sin(δ θ 1 ) - V 1V 2 2x 1 sin(θ 1 θ 2 ) (4.4) 0= -Re(V 2 θ 2 ( V 2 θ 2 Z L ) ) - V 1V 2 2x 1 sin(θ 2 θ 1 )- V 3V 2 x 2 sin(θ 2 θ 3 ) (4.5) 0 = - I m (V 2 θ 2 ( V 2 θ 2 ) ) - V 2 1 Z L 2x 1 + V 1V 2 cos(θ 2x 2 θ 1 ) - V 2 2 1 x 2 + V 3V 2 x 2 cos(θ 2 θ 3 ) (4.6) Fault DAE: δ =ω 0 ω (4.7) 2H ω = P m - E V 1 x d 0 = - V 1 2 x d sin(δ θ 1 )-K d ω (4.8) + V 1E cos(θ x 1 δ) - V 1 2 + V 1V 2 2x d 1 cos(θ 2x 1 θ 2 )- I m (V 1 θ 1 ( V 1 θ 1 ) )-Q 1 jx EE 1 (4.9) 36

0 = E V 1 x d sin(δ θ 1 ) - V 1V 2 2x 1 sin(θ 1 θ 2 ) - R e (V 1 θ 1 ( V 1 θ 1 jx 1 ) ) -P EE (4.10) 0= -Re(V 2 θ 2 ( V 2 θ 2 jx 1 Z L ) ) - V 1V 2 2x 1 sin(θ 2 θ 1 )- V 3V 2 x 2 sin(θ 2 θ 3 ) (4.11) 0 = - I m (V 2 θ 2 ( V 2 θ 2 ) ) - V 2 1 jx 1 Z L 2x 1 + V 1V 2 cos(θ 2x 2 θ 1 ) - V 2 2 1 x 2 + V 3V 2 x 2 cos(θ 2 θ 3 ) (4.12) P m = 1.0 P e = E V 1 x d sin(δ θ 1 ) (4.13) Post-fault DAE: δ =ω 0 ω (4.14) 2H ω = P m - E V 1 x d 0 = - V 1 2 x d sin(δ θ 1 ) (4.15) + V 1E cos(θ x 1 δ) - V 1 2 + V 1V 2 cos(θ d 2x 1 2x 1 θ 2 ) (4.16) 1 0 = E V 1 x d sin(δ θ 1 ) - V 1V 2 2x 1 sin(θ 1 θ 2 ) (4.17) 0= -Re(V 2 θ 2 ( V 2 θ 2 Z L ) ) - V 1V 2 2x 1 sin(θ 2 θ 1 )- V 3V 2 x 2 sin(θ 2 θ 3 ) (4.18) 0 = - I m (V 2 θ 2 ( V 2 θ 2 ) ) - V 2 1 Z L 2x 1 + V 1V 2 cos(θ 2x 2 θ 1 ) - V 2 2 1 x 2 + V 3V 2 x 2 cos(θ 2 θ 3 ) (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 0.00 0.42 0.48 0.55 0.2518 (CCT) 0.2508(CCT) 0.2498(CCT) (0.00%) (0.00%) (0.00%) 0.2576(CCT) 0.2566(CCT) 0.2555(CCT) 0.03 (2.30%) (2.31%) (2.28%) 0.2596(CCT) 0.2585(CCT) 0.2575(CCT) 0.04 (3.10%) (3.07%) (3.08%) 0.2616(CCT) 0.2605(CCT) 0.2595(CCT) 0.05 (3.89%) (3.87%) (3.88%) 0.2637(CCT) 0.2626(CCT) 0.2615(CCT) 0.06 (4.73%) (4.70%) (4.68%) 0.2657(CCT) 0.2646(CCT) 0.2635(CCT) 0.07 (5.52%) (5.50%) (5.48%) 0.2678(CCT) 0.2667(CCT) 0.2656(CCT) 0.08 (6.35%) (6.34%) (6.33%) 0.2699(CCT) 0.2688(CCT) 0.2677(CCT) 0.09 (7.19%) (7.18%) (7.17%) 0.2721(CCT) 0.2709(CCT) 0.2698(CCT) 0.10 (8.06%) (8.01%) (8.01%) 0.2955(CCT) 0.2940(CCT) 0.2927(CCT) 0.20 (17.36%) (17.22%) (17.17%) 0.3222(CCT) 0.3203(CCT) 0.3184(CCT) 0.30 (27.96%) (27.71%) (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 0.00 LOAD 0.64 0.75 0.92 0.2487 (CCT) 0.2478(CCT) 0.2468(CCT) (0.00%) (0.00%) (0.00%) 0.2544(CCT) 0.2534(CCT) 0.2524(CCT) 0.03 (2.29%) (2.26%) (2.27%) 0.2564(CCT) 0.2553(CCT) 0.2543(CCT) 0.04 (3.10%) (3.03%) (3.08%) 0.2583(CCT) 0.2605(CCT) 0.2562(CCT) 0.05 (3.86%) (5.13%) (3.81%) 0.2603(CCT) 0.2626(CCT) 0.2582(CCT) 0.06 (4.66%) (5.97%) (4.62%) 0.2624(CCT) 0.2646(CCT) 0.2602(CCT) 0.07 (5.51%) (5.50%) (5.43%) 0.2644(CCT) 0.2667(CCT) 0.2622(CCT) 0.08 (6.31%) (6.78%) (6.24%) 0.2665(CCT) 0.2688(CCT) 0.2642(CCT) 0.09 (7.16%) (7.63%) (7.05%) 0.2686(CCT) 0.2709(CCT) 0.2663(CCT) 0.10 (8.00%) (9.30%) (7.90%) 0.2912(CCT) 0.2940(CCT) 0.2884(CCT) 0.20 (17.09%) (18.64%) (17.17%) 0.3165(CCT) 0.3203(CCT) 0.3128(CCT) 0.30 (27.26%) (29.26%) (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 0.00 LOAD 1.16 1.57 2.39 0.2458(CCT) 0.2448(CCT) 0.2439(CCT) (0.00%) (0.00%) (0.00%) 0.2514(CCT) 0.2503(CCT) 0.2494(CCT) 0.03 (2.28%) (2.25%) (2.26%) 0.2533(CCT) 0.2522(CCT) 0.2513(CCT) 0.04 (3.05%) (3.02%) (3.03%) 0.2552(CCT) 0.2541(CCT) 0.2532(CCT) 0.05 (3.82%) (3.80%) (3.81%) 0.2572(CCT) 0.2561(CCT) 0.2551(CCT) 0.06 (4.64%) (4.62%) (4.59%) 0.2591(CCT) 0.2580(CCT) 0.2570(CCT) 0.07 (5.41%) (5.39%) (5.37%) 0.2611(CCT) 0.2600(CCT) 0.2590(CCT) 0.08 (6.22%) (6.21%) (6.19%) 0.2631(CCT) 0.2620(CCT) 0.2610(CCT) 0.09 (7.04%) (7.03%) (7.01%) 0.2652(CCT) 0.2641(CCT) 0.2630(CCT) 0.10 (7.89%) (7.88%) (7.83%) 0.2871(CCT) 0.2857(CCT) 0.2844(CCT) 0.20 (16.80%) (16.71%) (17.17%) 0.3165(CCT) 0.3203(CCT) 0.3075(CCT) 0.30 (28.76%) (30.84%) (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 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.20 0.30 0.2429(CCT) (0.00%) 0.2484(CCT) (2.26%) 0.2503(CCT) (3.05%) 0.2522(CCT) (3.83%) 0.2541(CCT) (4.61%) 0.2560(CCT) (5.39%) 0.2579(CCT) (6.18%) 0.2599(CCT) (7.00%) 0.2619(CCT) (7.82%) 0.2831(CCT) (16.55%) 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 0.00 LOAD 0.42 0.48 0.55 0.0299 (P=Q) 0.0261(P=Q) 0.0266(P=Q) 0.0307(P=Q) 0.0267(P=Q) 0.0231(P=Q) 0.03 0.0310(P=Q) 0.0269(P=Q) 0.0233(P=Q) 0.04 0.0311(P=Q) 0.0271(P=Q) 0.0234(P=Q) 0.05 0.0312(P=Q) 0.0274(P=Q) 0.0236(P=Q) 0.06 0.0314(P=Q) 0.0276(P=Q) 0.0238(P=Q) 0.07 0.0316(P=Q) 0.0278(P=Q) 0.0240(P=Q) 0.08 0.0317(P=Q) 0.0280(P=Q) 0.0242(P=Q) 0.09 0.0319(P=Q) 0.0283(P=Q) 0.0244(P=Q) 0.10 0.0339(P=Q) 0.0301(P=Q) 0.0265(P=Q) 0.20 0.0360(P=Q) 0.0318(P=Q) 0.0279(P=Q) 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.). 43

ES POWER 0.00 LOAD 0.64 0.75 0.92 0.0195 (P=Q) 0.0165(P=Q) 0.0135(P=Q) 0.0199(P=Q) 0.0168(P=Q) 0.0138(P=Q) 0.03 0.0200(P=Q) 0.0169(P=Q) 0.0138(P=Q) 0.04 0.0201(P=Q) 0.0170(P=Q) 0.0139(P=Q) 0.05 0.0202(P=Q) 0.0171(P=Q) 0.0140(P=Q) 0.06 0.0204(P=Q) 0.0172(P=Q) 0.0141(P=Q) 0.07 0.0205(P=Q) 0.0173(P=Q) 0.0142(P=Q) 0.08 0.0207(P=Q) 0.0174(P=Q) 0.0143(P=Q) 0.09 0.0208(P=Q) 0.0175(P=Q) 0.0114(P=Q) 0.10 0.0224(P=Q) 0.0187(P=Q) 0.0153(P=Q) 0.20 0.0240(P=Q) 0.0202(P=Q) 0.0163(P=Q) 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.). 44

ES POWER 0.00 LOAD L 1.16 1.57 2.39 0.0106 (P=Q) 0.0079(P=Q) 0.0052(P=Q) 0.0108(P=Q) 0.0080(P=Q) 0.0053(P=Q) 0.03 0.0109(P=Q) 0.0081(P=Q) 0.0053(P=Q) 0.04 0.0110(P=Q) 0.0081(P=Q) 0.0054(P=Q) 0.05 0.0111(P=Q) 0.0082(P=Q) 0.0054(P=Q) 0.06 0.0111(P=Q) 0.0082(P=Q) 0.0054(P=Q) 0.07 0.0112(P=Q) 0.0083(P=Q) 0.0054(P=Q) 0.08 0.0113(P=Q) 0.0084(P=Q) 0.0054(P=Q) 0.09 0.0113(P=Q) 0.0084(P=Q) 0.0055(P=Q) 0.10 0.0121(P=Q) 0.0089(P=Q) 0.0059(P=Q) 0.20 0.1280(P=Q) 0.0094(P=Q) 0.0062(P=Q) 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.). 45

ES POWER LOAD 4.84 0.00 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.20 0.30 0.0026(P=Q) 0.0026(P=Q) 0.0026(P=Q) 0.0026(P=Q) 0.0027(P=Q) 0.0027(P=Q) 0.0027(P=Q) 0.0027(P=Q) 0.0027(P=Q) 0.0029(P=Q) 0.0031(P=Q) 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 0.00 LOAD 0.42 0.48 0.55 7.5 KJ 6.5 KJ 5.6 KJ 7.9 KJ 6.9 KJ 5.9 KJ 0.03 8.0 KJ 7.0 KJ 6.0 KJ 0.04 8.1 KJ 7.1 KJ 6.1 KJ 0.05 8.2 KJ 7.2 KJ 6.2 KJ 0.06 8.3 KJ 7.3 KJ 6.3 KJ 0.07 8.5 KJ 7.4 KJ 6.4 KJ 0.08 8.6 KJ 7.5 KJ 6.5 KJ 0.09 8.7 KJ 7.7 KJ 6.6 KJ 0.10 10.0 KJ 8.8 KJ 7.8 KJ 0.20 11.6 KJ 10.2 KJ 8.9 KJ 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.). 47

ES POWER 0.00 LOAD 0.64 0.75 0.92 4.8 KJ 4.1 KJ 3.3 KJ 5.1 KJ 4.3 KJ 3.5 KJ 0.03 5.1 KJ 4.3 KJ 3.5 KJ 0.04 5.2 KJ 4.3 KJ 3.6 KJ 0.05 5.3 KJ 4.4 KJ 3.6 KJ 0.06 5.4 KJ 4.5 KJ 3.7 KJ 0.07 5.4 KJ 4.6 KJ 3.7 KJ 0.08 5.5 KJ 4.6 KJ 3.8 KJ 0.09 5.6 KJ 4.7 KJ 3.8 KJ 0.10 6.5 KJ 5.4 KJ 4.4 KJ 0.20 7.6 KJ 6.4 KJ 5.1 KJ 0.30 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 0.00 LOAD 1.16 1.57 2.39 2.6 KJ 1.9 KJ 1.3 KJ 2.7 KJ 2.0 KJ 1.3 KJ 0.03 2.8 KJ 2.0 KJ 1.3 KJ 0.04 2.8 KJ 2.1 KJ 1.4 KJ 0.05 2.9 KJ 2.1 KJ 1.4 KJ 0.06 2.9 KJ 2.1 KJ 1.4 KJ 0.07 2.9 KJ 2.2 KJ 1.4 KJ 0.08 3.0 KJ 2.2 KJ 1.4 KJ 0.09 3.0 KJ 2.2 KJ 1.4 KJ 0.10 3.5 KJ 2.5 KJ 1.7 KJ 0.20 4.0 KJ 2.9 KJ 1.9 KJ 0.30 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 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.20 0.30 0.6 KJ 0.6 KJ 0.7 KJ 0.7 KJ 0.7 KJ 0.7 KJ 0.7 KJ 0.7 KJ 0.7 KJ 0.8 KJ 0.9 KJ 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 0.00 LOAD 0.42 0.48 0.55 0.0 KJ 0.0 KJ 0.0 KJ 7.7 KJ 7.7 KJ 7.7 KJ 0.03 10.4 KJ 10.3 KJ 10.3 KJ 0.04 13.1 KJ 13.0 KJ 13.0 KJ 0.05 15.8 KJ 15.8 KJ 15.7 KJ 0.06 18.6 KJ 18.5 KJ 18.4 KJ 0.07 21.4 KJ 21.3 KJ 21.2 KJ 0.08 24.3 KJ 24.2 KJ 24.1 KJ 0.09 27.2 KJ 27.1 KJ 27.0 KJ 0.10 59.1 KJ 58.8 KJ 58.5 KJ 0.20 96.7 KJ 96.1 KJ 95.5 KJ 0.30 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 0.00 LOAD 0.64 0.75 0.92 0.0 KJ 0.0 KJ 0.0 KJ 7.6 KJ 7.6 KJ 7.6 KJ 0.03 10.3 KJ 10.2 KJ 10.2 KJ 0.04 12.9 KJ 12.9 KJ 12.8 KJ 0.05 15.6 KJ 15.6 KJ 15.5 KJ 0.06 18.4 KJ 18.3 KJ 18.2 KJ 0.07 21.2 KJ 21.1 KJ 21.0 KJ 0.08 24.0 KJ 23.9 KJ 23.8 KJ 0.09 26.9 KJ 26.8 KJ 26.6 KJ 0.10 58.2 KJ 58.0 KJ 57.7 KJ 0.20 94.9 KJ 94.4 KJ 93.8 KJ 0.30 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 0.00 LOAD 1.16 1.57 2.39 0.00 KJ 0.00 KJ 0.00 KJ 7.6 KJ 7.5 KJ 7.5 KJ 0.03 10.1 KJ 10.1 KJ 10.1 KJ 0.04 12.8 KJ 12.7 KJ 12.7 KJ 0.05 15.4 KJ 15.4 KJ 15.3 KJ 0.06 18.1 KJ 18.1 KJ 18.0 KJ 0.07 20.9 KJ 20.8 KJ 20.7 KJ 0.08 23.7 KJ 23.6 KJ 23.5 KJ 0.09 26.5 KJ 26.4 KJ 26.3 KJ 0.10 57.4 KJ 57.1 KJ 56.9 KJ 0.20 93.3 KJ 92.8 KJ 92.2 KJ 0.30 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 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.20 0.30 0.00 KJ 7.50 KJ 10.0 KJ 12.6 KJ 15.2 KJ 17.9 KJ 20.6 KJ 23.4 KJ 26.2 KJ 56.6 KJ 91.7 KJ 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.). 55

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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