report on first swing stability

First Swing Stability

TABLE OF CONTENT

1) Chapter 1

Overview of power oscillations& stability ……………………………………………………….……..……..3

SECTION I : Power Angle Equation---------------------------------------------------6 SECTION II : Swing Equation-----------------------------------------------------------8

SECTION III : Transmission Line Analysis-------------------------------------------9

SECTION IV : Equal Area Criteria-----------------------------------------------------10

2) Chapter 2

First Swing Stability……………………………………………………………………………………..…................14

First Swing Stable-----------------------------------------------------------------------15 Rotor Angle Stability-------------------------------------------------------------------17 Stability Margin of Single Machine System---------------------------------------18 Multi- machine stability---------------------------------------------------------------21

3) Chapter 3

Improvement of FSS using FACTS devices………………………………………………………………………….24

SECTION I : by ideal Shunt capacitor------------------------------------------------25 SECTION II : by shunt FACT devices--------------------------------------------------28

Using SVC-----------------------------------------------------------------------------30 SCETION III : By STATCOM-------------------------------------------------------------33

Conclusion…………………………………………………………………….……………………………………35References…………………………………………………………………………………………………………36

NATIONAL INSTITUTE OF TECHNOLOGY HAMIRPUR (H.P.) Page 1


CHAPTER 1

Overview of power oscillations

&

Stability



INTRODUCTION

The first electric power system was a dc system built by Edison in 1882. The subsequent power systems that were constructed in the late 19 th century were all dc systems. However despite the initial popularity of dc systems by the turn of the 20th century ac systems started to outnumber them. The ac systems were thought to be superior as ac machines were cheaper than their dc counterparts and more importantly ac voltages are easily transformable from one level to other using transformers. The early stability problems of ac systems were experienced in 1920 when insufficient damping caused spontaneous oscillations or hunting. These problems were solved using generator damper winding and the use of turbine-type prime movers.

The stability of a system refers to the ability of a system to return back to its steady state when subjected to a disturbance. As mentioned before, power is generated by synchronous generators that operate in synchronism with the rest of the system. A generator is synchronized with a bus when both of them have same frequency, voltage and phase sequence. We can thus define the power system stability as the ability of the power system to return to steady state without losing synchronism.

Usually power system stability is categorized into following categories:

1) Rotor Angle Stability

2) Frequency Stability

3) Voltage Stability

In this chapter we shall discuss the transient stability aspect of a power system.



Fig 1:Classifications of power system stability

Types of Synchronous stability:

Synchronous stability may be divided in to two main categories depending upon magnitude of the disturbance.

1) Steady state stability2) Transient stability

1) Steady state stability: The steady state stability is the ability of a system to bring it to a stable condition after a small disturbance. The study of a steady state stability is basically concerned with the effect of gradual infinitesimal power changes.

Types of Steady state stability:

Steady state stability is subdivided to make a distinction between operations with and without automatic control devices such as governors and voltage regulators.

i) Static stability: Static stability refers to inherent stability that prevails without the aid of automatic control devices.



ii) Dynamic stability: Dynamic stability denotes artificial stability given to an inherently unstable system by automatic control devices. Dynamic stability is concerned with small disturbances lasting for the times of the order of 10 to 30 seconds.

2) Transient stability: The transient stability is the ability of the system to bring it to a stable condition after a large disturbance. Transient stability is concerned with the sudden and large changes in the network conditions. The large disturbances can occur due to sudden changes in application or removal of loads, line switching operating operations, line faults or loss of excitation.

Stability limits:

The stability limit is the maximum power that can be transferred in a network between sources and loads without loss of synchronism.

Types of stability limits:

1) Steady state limit: The steady state limit is the maximum power that can be transformed without the system becoming unstable when the load is increased gradually under steady state conditions.

2) Transient Limit: Transient limit is the maximum power that can be transformed without the system becoming unstable when a sudden or large disturbance occurs.

The system experiences a shock by sudden and large power changes and violent fluctuations of voltage occur. Consequently, individual machines or group of machines may go out of step. The rapidity of the application of a large disturbances is responsible for the loss of stability, it may be possible to maintain stability if the same large load is applied gradually. Thus the transient stability limit is lower than the steady state limit.

NOTE:- First Swing Stability is also known as Rotor Angle Stability and

Electromechanical Oscillations



Section I : Power Angle Equation:

The expression establishing the relationship between the active power transferred (Pe) to the system and the angle δ is known as power angle equation.

The expression for the active power transferred to the system is given by

Pe = Sin δ ----------------- (1)

Where:

X = Xd + Xl

X ―› transfer reactance

Xd ―› synchronous / transient reactance of the machine

Xl ―› reactance of the transmission line

E ―› Magnitude of the voltage behind direct axis synchronous reactance of the machine

V ―› Voltage of infinite bus

δ ―› Angle between the voltages E and V

The maximum steady state power transfer occurs when δ = 90⁰. From equation (1),

Pe max = Sin 90⁰ =

Pe = Pe max Sin δ

Power – Angle curve: The graphical representation of power Pe and the load angle δ is called the power angle diagram or power-angle curve.

Maximum power is transferred when δ=90⁰. As δ is increased beyond 90⁰, Pe

decreases and become zero at δ = 180⁰, Pe becomes negative which implies that the power flow direction is reversed and the power is supplied from the infinite bus to the generator. The value of Pe max is often called the pull-out power. It is also called the steady state limit.

For transient conditions, the transient reactance X´d is used.



Transient Stability:

The following simplifying assumptions are made in the study of transient stability.

(a) System resistances may be neglected in comparison with reactance.(b)The machine has cylindrical rotor. The direct-axis reactance (Xd) is equal

to the quadrature -axis reactance (Xq).(c) The system may be assumed to supply an infinite bus.(d)Each machine may be assumed to supply an infinite bus.(e) Direct axis transient reactance (Xd) is used for machine representation.(f) The shaft input power may be assumed constant for few seconds after

occurrence of a disturbance. This assumption may be valid on the grounds that the mechanical system involving governors, steam valves etc. are relatively sluggish in operation as compared to rapidly changing electrical quantities. With fast acting valves the assumption of constant input will not be true.

In a synchronous generator the input is the mechanical or shaft torque and the output is the electromagnetic torque. Both these torques are assumed positive in the following discussion. For a synchronous motor, the input is the electromagnetic torque and the output is the shaft torque. Based on the sign conventions adopted for synchronous generators, the values of the shaft torque and electromagnetic torque are taken as negative for motor action.

Let Te→ Electromagnetic torque

Ts→ Shaft torque

If losses are neglected the difference between the shaft torque and the electromagnetic torque is equal to the accelerating or decelerating torque.

Ts – Te = Ta

Where Ta → Accelerating torque

For a generator:

i) When Ts > Te , Then Ta is positive and rotor accelerates



ii) When Ts < Te, then Ta is negative and rotor decelerates

For a motor:

i) When Ts < Te , Then Ta is positive and rotor accelerates

ii) When Ts > Te , then Ta is negative and rotor decelerates

Section II: Swing Equation:

The equation establishing the relationship between the accelerating power and angular acceleration is called swing equation. It is a non-linear differential equation of the second order.

M = Ps – Pe = Pa

M = Jω

θ = ωst + δ

Where

M→ angular momentum of the rotor

J → moment of inertia of the rotor

ω → synchronous speed of the rotor

Ps → mechanical power input

Pe→ electrical power output

Pa → accelerating power

θ → angular position of the rotor with respect to reference axis at any instant t

Swing Curve:

Graphical representation of δ (usually in electrical radians) and time t (in seconds) is called the swing curve. Swing curve provide information regarding stability.



If δ increases continuously with time the system is unstable. While if δ starts decreasing after reaching a maximum value it is inferred that the system will remain stable.

Fig 2: Swing Curve

Swing curves are useful in determining the adequacy of relay protection on power systems with regard to the clearing of faults before one or more machines become unstable and fall out of synchronism. The critical clearing time is found to specify the correct speed of the circuit breaker.

The solution of the swing equation involves elliptic integrals. Step-by-step (or point-by-point) may be used for numerical solution of swing equation. At present digital computer is used for solving swing equation.

Section III: Transmission line analysis:

A transmission line possesses resistance R, inductance L, capacitance C and shunt or leakage conductance G. All the parameters are distributed along the line.



All low-voltage overhead lines having length up to 80 km are generally categorized as short lines. For overhead lines up to 80 km the capacitance C and shunt conductance G is negligibly small. So these types of transmission lines are generally R-L type.

For a R-L type transmission line the time constant (τ) is given by the ratio of L and R.

τ =

The damping factor is the reciprocal time constant (τ).

Damping Ratio = =

Now we know that for the quick damped out oscillations the damping ratio should be high (it means time constant should be less). So to make the damping ratio high we can make two things

1) Either we can increase the resistance (R) of the transmission line.2) Or we can decrease the value of inductance (L) during only the fault

condition.

Section IV: Equal Area Criterion

The real power transmitted over a lossless line is given by (9.4). Now consider the situation in which the synchronous machine is operating in steady state delivering a power Pe equal to Pm when there is a fault occurs in the system. Opening up of the circuit breakers in the faulted section subsequently clears the fault. The circuit breakers take about 5/6 cycles to open and the subsequent post-fault transient last for another few cycles. The input power, on the other hand, is supplied by a prime mover that is usually driven by a steam turbine. The time constant of the turbine mass system is of the order of few seconds, while the electrical system time constant is in milliseconds. Therefore, for all practical purpose, the mechanical power is remains constant during this period when the electrical transients occur. The transient stability study therefore concentrates on the ability of the power system to recover from the fault and deliver the constant power Pm with a possible new load angle δ .



Consider the power angle curve shown in Fig. 9.3. Suppose the system of Fig. 9.1 is operating in the steady state delivering a power of Pm at an angle of δ0

when due to malfunction of the line, circuit breakers open reducing the real power transferred to zero. Since Pm remains constant, the accelerating power Pa

becomes equal to Pm . The difference in the power gives rise to the rate of change of stored kinetic energy in the rotor masses. Thus the rotor will accelerate under the constant influence of non-zero accelerating power and hence the load angle will increase. Now suppose the circuit breaker re-closes at an angle δc. The power will then revert back to the normal operating curve. At that point, the electrical power will be more than the mechanical power and the accelerating power will be negative. This will cause the machine decelerate. However, due to the inertia of the rotor masses, the load angle will still keep on increasing. The increase in this angle may eventually stop and the rotor may start decelerating, otherwise the system will lose synchronism.

Note that

Fig.3: Power-angle curve for equal area criterion.

Now suppose the generator is at rest at δ0. We then have dδ / dt = 0. Once a



fault occurs, the machine starts accelerating. Once the fault is cleared, the machine keeps on accelerating before it reaches its peak at δc , at which point we again have dδ / dt = 0.

the area of acceleration is given by A1 while the area of deceleration is given by A2.

Now consider the case when the line is reclosed at δc such that the area of acceleration is larger than the area of deceleration, i.e., A1 > A2 . The generator load angle will then cross the point δm , beyond which the electrical power will be less than the mechanical power forcing the accelerating power to be positive. The generator will therefore start accelerating before is slows down completely and will eventually become unstable. If, on the other hand, A1 < A2 , i.e., the decelerating area is larger than the accelerating area, the machine will decelerate completely before accelerating again. The rotor inertia will force the subsequent acceleration and deceleration areas to be smaller than the first ones and the machine will eventually attain the steady state. If the two areas are equal, i.e., A1 = A2 , then the accelerating area is equal to decelerating area and this is defines the boundary of the stability limit. The clearing angle δc

for this mode is called the Critical Clearing Angle and is denoted by δcr. We then get from Fig.3 by substituting δc = δcr

Since the critical clearing angle depends on the equality of the areas, this is called the equal area criterion.

Example 9.3:

Consider the system of Example 9.1. Let us assume that the system is operating with Pm = Pe = 0.9 per unit when a circuit breaker opens inadvertently isolating the generator from the infinite bus. During this period the real power transferred becomes zero. From Example 9.1 we have calculated δ0 = 23.96 ° = 0.4182 rad and the maximum



power transferred as

per unit

We have to find the critical clearing angle.

From (9.15) the accelerating area is computed as by note that Pe = 0 during this time. This is then given by

To calculate the decelerating area we note that δm = π - 0.4182 = 2.7234 rad. This area is computed by noting that Pe = 2.2164 sin(δ ) during this time. Therefore

Equating A1 = A2 and rearranging we get



CHAPTER 2




First swing stable

Power systems are becoming more complex because of the increase in interconnection for economic operation, better reliability and strategic coverage against catastrophic outages. The transmission networks are now under more stress than ever before to avoid the capital cost involved in reinforcement and environmental objections. These trends have resulted in the need to operate a system closer to the stability limits, and thus the system becomes more vulnerable to disturbances. Power utilities increasingly face the threat of transient and dynamic stability problems. Utility engineers perform a huge number of off-line transient stability simulations to determine the operating security limits. These limits are then used in the energy management system at the control centre for on-line dynamic security monitoring.

Definition:

A power system is said to be first-swing stable if the post-halt angle, in the centre of inertia (COI) reference frame, of all severely disturbed machines (SDM) initially increases (or decreases) until a peak value is reached where the angle starts returning to the stable equilibrium point. Existence of peak angle, and hence zero speed, of all SDMs guarantees the first swing stability of the system. On the other hand, the system is considered to be first swing unstable if the post-fault angle of at least one of the machines in the system increases (or decreases) monotonically and eventually becomes unbounded (exceeds 180" in the CO1 reference frame). The first swing stability of a machine can also be checked by observing the variation of machine speed and accelerating power Pa in the post-fault period. A stable machine reaches the peak angle (or zero speed) in the post-fault period while its accelerating power, and hence acceleration, is still negative:



Note that the accelerating power Pa and acceleration have the same sign and they are related through the machine inertia constant. A machine is considered

to be unstable if its angle continues to increase ( > 0) when its accelerating power changes sign (or crosses theZero value):

The critical situation of a machine is characterized by the simultaneous occurrence of zero speed and accelerating power in the post-fault period:

It may be mentioned here that the above criteria are valid for a machine that has a tendency to run out of step by acceleration. For a decelerating machine, the above criteria are to be modified by adding a negative sign to the left-hand side of eqns by comparing the criterion for the stable and critical situations, the negated machine accelerating power at zero speed may be considered as an index for the degree of stability of the machine. Similarly, by comparing the criterion for the unstable and critical situations, the machine speed at zero accelerating power may be considered as an index for the degree of instability of the machine. Thus determination of the degree of stability/ instability of a machine requires the machine speeds and accelerating powers in the post-fault period.

First Swing StabilityFaults (short circuits) in the power system cause very fast changes in the electrical conditions. The changed electrical state influences electrical power output from generators, changes in power flows and in load demand.

Generators will receive almost the same mechanical input through the shaft during the fault as before the fault. This will cause a power imbalance between mechanical input and electrical output, which will accelerate or decelerate each individual rotor with respect to the rest of the system. The individual rate of change is determined by the power deviation and the rotor



inertia of each generator. The rotor angle differences in the power system will increase and this angle difference cannot be too large. If this happens the generator fall out of step, grid node voltages will become zero in certain parts of the grid, some generators will start to work as motors etc.

Therefore, when the fault is cleared (after roughly 0.1 s) the power system has to be restored to sufficiently small angle deviations between the generator rotors again. After the first swing we require damping of the oscillations.

Rotor Angle Stability (Electromechanical Oscillations)

All sudden changes in a power system are associated with a number of phenomena with different timeframes involved. In the first phase, the electrical properties are very quickly adjusting to the new situation. This changes the share of power production between different generators and it also causes changes in load demand. The power flows in the grid changes accordingly.

In the second phase the unbalance between mechanical input and electrical output of each generator are causing a change of generator mechanical speed. The individual rate of change in speed is decided by the power deviation and the rotor inertia. When generators are changing speed with different rates will the rotor angles of each generator start to deviate from the predisturbance value. This causes a change in power flows in the grid causing further imbalance for each generator.

In the third phase protection and control are coming into play. Any faults are disconnected - usually after a short time delay (associated with the problems of breaking high currents). The fault disconnection causes a new transient. Controls are trying to restore the grid to steady state conditions again. They operate with different speeds depending on what they control. The voltage regulators tries to restore voltage and turbine governors adjust mechanical input to generators so we return to balance between consumption and production again.

These transitions are oscillatory in its nature and very lightly damped. The phenomenon is usually called Rotor Angle Stability or Electromechanical Power Oscillations. Of particular interest is the so-called first swing stability, which indicates that the generators do not swing too far from each other on the 1st-oscillation.

Rotor angle oscillations can also arise in the grid without any obvious reason. High power flows over weak transmission lines, fast and powerful voltage


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regulators and other types of controls may cause standing oscillations in the grid. Close located generators tend to attach to each other and the system can end up with machine groups oscillating towards each other with very low frequency (down to 0.1-0.2 Hz). The tie lines can then become heavily loaded if many generators oscillate towards another group.

It is important to damp these oscillations as quickly as possible. They causes mechanical wear in power plants may cause power quality problems (flicker, etc.). The system is also more vulnerable if further disturbances occur. Two main ways can be identified. The first one is to influence the transmission capacity and as a second option try to inject or extract active power into or out from the electrical grid that oscillates.

Stability margin of a single machine system

The objective of this Section is to demonstrate the proposed method of determining the SM of a simple single machine infinite bus (SMIB) system as shown in Fig. 4.

The procedure for determining the SM for various faults clearing times (both for stable and unstable situations) is described in the following sections.

Stable situation



Fig. 5 shows the variation of machine angle and speed for a stable situation. It can be noticed in the Figure that the machine angle δ increases in the early part of the post-fault period until it reaches the peak value δp, at time tp. While the machine speed ω decreases in the post-fault period and crosses the zero value (or changes sign) at the same time tp because ω is the time derivative of δ . Existence of peak angle or zero speed guarantees the first swing stability of the machine.



The well known 'equal area criterion' of the system is shown in Fig. 6. The locus of the power angle curve, from 0 to t,,, is

Here δ0 and δc are the pre-fault and fault clearing angles, respectively. The stability criterion A, = A, is satisfied when the locus reaches the point c. At c (in Fig. 3), the machine angle reaches the peak value δp and the corresponding time is tu (in Fig. 2). The intercept cd (in Fig. 3) represents the decelerating power (or negative accelerating power) of the machine at tp. The decelerating power at zero speed pushes the machine towards the post fault stable equilibrium point δs As described in Section 2, the decelerating power or intercept cd may be considered as the degree of stability of the machine. Thus the SM of the machine is:

When tcl , l (and hence: δcl )increases, a,, approaches to the unstable equilibrium point δu, and the SM (or intercept,ea') decreases. When tcl = tcr, δp coincides with δu, and the SM becomes zero. If tcl<<tcrthe peak angle δp may



occur before reaching the maximum decelerating power point (intercept xy at δ=180/2 in Fig. 6). In this case, the system is considered as 'too stable' and the SM is not determined from eqn. 8 because it may provide a wrong apprehension. That is the SM may increase with the increase of tcl.

Unstable situation

the machine angle increases monotonically and the speed cannot reach the zero value in the post fault period. Fig. 4 shows the variation of machine speed for two different fault clearing times (both for unstable situations). It can be noticed in the Figure that for tcl = f a (= O.l s), the machine speed initially decreases in the post-fault period and reaches a minimum value

of then starts increasing. Note that the machine accelerating power changes sign when the speed reaches the minimum value. As mentioned earner, the machine speed, when Pa = 0, can be considered as a measure of the degree of instability of the machine. The corresponding kinetic energy can also represent the degree of instability of the machine. Thus the instability margin (IM) of the machine can be written as

When the machine speed may increase even in the early part of the post-fault period as shown in Fig. 4 for tcl = tb ( = 0.32s). In this case, the zero accelerating power point is crossed before the fault clearing time and the system is considered as 'too unstable'. For such a situation, the minimum speed in the post-fault period is the same as the speed at fault clearing time and the corresponding IM can be considered as

Above equation is an approximate representation of the degree of instability of the machine for a 'too unstable' situation and it increases with the increase of fault clearing time.



Multimachine Stability

Oscillations in s Two Area System

Consider Fig. 8, which depicts a number of weights that are suspended by elastic strings. The weights represent generators and the electric transmission lines being represented by the strings. Note that in a transmission system, each transmission line is loaded below its static stability limit. Similarly, when the mechanical system is in static steady state, each string is loaded below its break point. At this point one of the strings is suddenly cut. This will result in transient oscillations in the coupled strings and all the weights will wobble. In the best possible case, this may result in the coupled system settling down to a


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new steady state. On the other hand, in the worst possible scenario this may result in the breaking of one more additional string, resulting in a chain reaction in which more strings may break forcing a system collapse. In a similar way, in an interconnected electric power network, the tripping of a transmission line may cause a catastrophic failure in which a large number of generators are lost forcing a blackout in a large area.

Fig. 8

Modern power systems are interconnected and operate close to their transient and steady state stability limits. In large interconnected systems, it is common to find a natural response of a group of closely coupled machines oscillating against other groups of machines. These oscillations have a frequency range of 0.1 Hz to 0.8 Hz. The lowest frequency mode involves all generators of the system. This oscillation groups the system into two parts - with generators in one part oscillating against those of the the other part. The higher frequency modes are usually localized with small groups oscillating against each other. Unfortunately, the inter-area oscillation can be initiated by a small disturbance in any part of the system. These small frequency oscillations fall under the category of dynamic stability and are analysed in linear domain through the liberalisation of the entire interconnected systems model.

Oscillations in a Two Area System

Consider the simple power system shown in Fig. 9 in which two machines are operating. Let us assume that starting with the initial angles δ1 and δ2 with respect to some reference at nominal frequency, machine 1 accelerates while machine 2 decelerates from this nominal frequency. We then have

(9.1)



where the subscripts 1 and 2 refer to machines 1 and 2 respectively. Let us assume that the transmission line is loss less. Then in the simple case where the power from machine 1 flows to machine 2, we get

(9.2)

Where δ12 = δ1 - δ2 .

Fig. 9: Single-line diagram of a two-machine power system.

Now since the system is lossless, (9.2) will also imply that Pm1 = - Pm2 . This means that in the steady state, the power generated at machine 1 is absorbed through machine 2. Combining (9.1) and (9.2) we get

………………(9.3)

Let us now assume that H1 = H2 = H , V1 = V2 = 1.0 per unit and Pm1 = 0

where the oscillation frequency ω is given by:

(9.4)



Thus the weighted difference of angles will approximate simple harmonic motion for small changes in δ12 and the frequency will decrease for an increase in inertia H or impedance X . Another aspect can be seen by adding the system to give

(9.5)

Thus the overall acceleration of the machine group will depend on the overall balance between power generated and consumed. Usually there are governors on the generators to reduce generated power if the system frequency increases.



CHAPTER 3

Improvement of FSS

using

FACTS Devices

Improvement of First Swing Stability Limit



Section I: By Ideal Shunt Compensator

Improving Voltage Profile Improving Power-Angle Characteristics

Improving Stability Margin

Improving Damping to Power Oscillations

The ideal shunt compensator is an ideal current source. We call this an ideal shunt compensator because we assume that it only supplies reactive power and no real power to the system. It is needless to say that this assumption is not valid for practical systems. However, for an introduction, the assumption is more than adequate. We shall investigate the behavior of the compensator when connected in the middle of a transmission line. This is shown in Fig. 10.1, where the shunt compensator, represented by an ideal current source, is placed in the middle of a lossless transmission line. We shall demonstrate that such a configuration improves the four points that are mentioned above.

Fig 10 Schematic diagram of an ideal, midpoint shunt compensation

Improving Stability Margin

This is a consequence of the improvement in the power angle characteristics and is one of the major benefits of using midpoint shunt compensation. As mentioned before, the stability margin of the system pertains to the regions of acceleration and deceleration in the power-angle curve. We shall use this concept to delineate the advantage of mid point shunt compensation.

Consider the power angle curves shown in Fig. 11.







Fig. 11 (a) is for an uncompensated system (b) for the compensated system.

Both these curves are drawn assuming that the base power is V2/X . Let us assume that the uncompensated system is operating on steady state delivering

an electrical power equal to Pm with a load angle of δ0 when a three-phase fault occurs that forces the real power to zero. To obtain the critical clearing angle for

the uncompensated system is δcr , we equate the accelerating area A1 with the decelerating area A2 , where

with δmax = π - δ0 . Equating the areas we obtain the value of δcr as

10.11

Let us now consider that the midpoint shunt compensated system is working with the same mechanical power input Pm. The operating angle in this case is δ1

and the maximum power that can be transferred in this case is 2 per unit. Let the fault be cleared at the same clearing angle δcr as before. Then equating areas A3

and A4 in Fig. 10.6 (b) we get δ2, where



Section II: By Shunt FACTS Devices

TRANSIENT stability is the main factor that limits the power transfer capability of long distance transmission lines. Power utilities are now placing more emphasis on improving the transient stability, especially the first swing stability limit, to increase the utilization of existing transmission facilities. A power system can be considered as first swing stable if the post-fault angle of all machines in center of angle (COA) reference frame increases (decreases) until a peak (valley) is reached when the angle starts returning . In other words, existence of zero speed (maximum or minimum angle) of all machines guarantees the first swing stability of the system. In general, a first swing stable system is considered as stable because system damping, governor, etc. usually help to damp oscillation in subsequent swings .The first swing stability limit of a single machine infinite bus (SMIB) system can be determined through equal area criterion (EAC) that depends on the difference between input mechanical power and output electrical power of the machine. During faulted period, the output power of the machine reduces dras- tically while the input mechanical power remains more or less constant and thus the machine accelerates. The turbine delivers excess energy to the machine and that can be represented by an area called accelerating area. To maintain the first swing stability, the machine must transfer the excess energy to the network once the fault is cleared. The excess energy transferring capability of the machine can be represented by another area called decelerating area and it depends on post-fault network condition. The stability limit can be improved by enlarging the decelerating area in early part of post-fault period. Initially, it was considered that the network condition cannot be controlled fast enough to enlarge the decelerating area dynamically. How- ever, recent development of power electronics introduces the use of flexible ac transmission system (FACTS) devices in power systems. FACTS devices are capable of controlling the network condition in a very fast manner and this unique feature of FACTS devices can be exploited to enlarge the decelerating area and hence improving the first swing stability limit of a system. Static VAR compensators (SVC) and static synchronous compensators (STATCOM) are members of FACTS family that are connected in shunt with the system. Even



though the primary purpose of shunt FACTS devices is to support bus voltage by injecting (or absorbing) reactive power, they are also capable of improving the transient stability and damping of a power system. The stability or damping can be improved by increasing (decreasing) the power transfer capability when the machine angle increases (decreases) and this can be achieved by operating the shunt FACTS devices in capacitive (inductive) mode

Continuous and discontinuous types of control are very commonly used for shunt FACTS devices to improve the transient stability and damping of a power system. The continuous control may not utilize the full capability of the device. On the other hand, the discontinuous control operates the device at its full rating to provide the maximum benefit. The continuous control is found to be very effective in improving the dynamic stability problem caused by small disturbances. However, to improve the transient stability, much larger control action is needed and it is suggested that the discontinuous control (also called bang-bang control, or BBC) should be used for this purpose. In BBC, the mode of operation of the device is changed (from full capacitive to full inductive or vice versa) at some discrete points. Usually the machine speed signal is used to change the mode of operation, but any signal that is dynamically related to machine speed can also be used. References used some locally measured signals to estimate the machine angle and speed of a simple radial system. However, the same techniques may not be applied to a general multi-machine system. The BBC maximizes the power transfer capability or decelerating area by operating the shunt FACTS devices at full capacitive rating. However, it is found in this study that, the speed based BBC is unable to utilize the entire decelerating area in improving the first swing stability limit. In fact, the use of last portion of decelerating area causes chattering action and that may eventually lead to instability. Such a situation occurs when the fault clearing time tc approaches the actual critical clearing time tcr .



Fig.10.A SMIB system with a shunt FACTS device: (a )single line diagram

Fig.11 .A SMIB system with a SVC.

Consider a lossless SMIB system with a shunt FACTS device as shown in Fig. 10(a). The equivalent circuit of the system is shown in Fig. 10(b) where E’ and V represent the machine internal voltage and infinite bus voltage, respectively. X1 is the reactance between bus m and the machine internal bus, and X2

is the reactance between bus m and the infinite bus. The mechanism of improving the first swing stability limit by utilizing the full benefit of shunt FACTS devices (SVC and STATCOM) is described in the following.

A. SVC



A SVC can be modelled by a variable shunt susceptance BSVC as shown in Fig. 11(a) . For a given BSVC , the transfer reactance X12 in Fig. 11(b) can be written as

The electrical output power of the machine in Fig. 11(b) is

The power-angle curve of the system with a SVC is shown in Fig. 12(a). When a fault occurs, Pe suddenly decreases from point a to b point and thus the machine starts accelerating along b-c where both ω and Pa are positive. At fault clearing, Pe suddenly increases and the area a-b-c-d-a represents the accelerating area Aa

If the SVC operates in capacitive mode (at fault clearing), Pe increases to point e where ω>0 and Pa<0. Thus the machine starts decelerating but its angle continues increasing along e-f until reaches a maximum value δm at point f , for a stable situation. The area represents the decelerating area e-f-g-d-eand it must be the same as Aa. The unused decelerating area f-h-g-f is a measure of stability margin (SM). Note that both and SM can be increased by raising the power curve as much as possible and which can be achieved by operating the SVC at its full capacitive rating.



Fig.12.P- δ curves for various operating conditions of SVC: (a) areas when the angle increases and (b) areas when the angle decreases.



For critically stable situation , almost the entire unused decelerating area is needed to counterbalance the accelerating area. In this case, the maximum angle occurs almost at point h . When the BBC changes the mode of operation of the SVC (from capacitive to inductive) at point h (or when ω=0),Pe suddenly decreases to point k and that makes the system first swing unstable. If the maximum angle occurs before reaching the point m , switching the operation of SVC to inductive mode (at ω=0 ) would not allow to increase the angle further. Such a situation occurs when tc is signifcantly less than tcr. However, the maximum angle occurs in between m and h , the BBC decreases Pe below Pm and that may not guarantee the first swing stability of the system. In other words, the BBC is unable to utilize the last portion of decelerating area (area m-h-n-m)to counterbalance the accelerating area.

Section III: By STATCOM

A STATCOM can be represented by a shunt current source as shown in Fig. 13. The STATCOM current is always in quadrature with its terminal voltage and can be written as (for capacitive mode of operation)



Fig13 : A system with STATCOM with equivalent circuit

The voltage magnitude and angle of bus are given by

-------------1

-------------2

For an inductive mode of operation, ISTAT in (1) and (2) is to be replaced by (-ISTAT) . The electrical output power of the machine in Fig. 13 can be written as

-------------3



The P- δ curve of the system with a STATCOM is shown in Fig. 14. For capacitive mode of operation(ISTAT>0) , the P- δ curve is not only raised but also shifted toward right (see Fig. 13) and that provides more decelerating area and hence higher stability limit. Similar to SVC, the control strategy of STATCOM that maximizes the rst swing stability limit and improves damping in subsequent swings can be considered as:



Conclusion

When the fault is cleared (after roughly 0.1 s) the power system has to be restored to sufficiently small angle deviations between the generator rotors again. After the first swing we require damping of the oscillations.

In the first phase, the electrical properties are very quickly adjusting to the new situation.

In the second phase the unbalance between mechanical input and electrical output of each generator are causing a change of generator mechanical speed.

In the third phase protection and control are coming into play.

FACTS controller contribute to stable the first swing as if the system remains stable during first swing it will promises the further stability of the system.



Refrences

1. J.J. Ford, G. Ledwi ch, Z.Y. Dong: Efficient and robust model predictive control for first swing transient stability of power systems using flexible AC transmission systems devices). IEEE 26th September 2007.

2. HAQUE M.H.: ‘Improvement of first swing stability limit by utilising full benefits of shunt FACTS devices’, IEEE Trans. Power Syst., 2004, 19, (4), pp. 1894–1902.

3. Prabha Kundur (Canada, Convener), John Paserba (USA, Secretary), Venkat Ajjarapu (USA), Gran Andersson (Switzerland), Anjan Bose (USA) , Claudio Canizares (Canada), Nikos Hatziargyriou (Greece), David Hill(Australia), Alex Stankovic (USA), Carson Taylor (USA), Thierry Van Cutsem (Belgium), and Vijay Vittal (USA): Definition and Classification of Power System Stability, IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 2, MAY 2004.

4. M.H. Haque: ‘Novel method of finding the first swing stability margin of a power system from time domain simulation’, IEE Pvoc.-Genev. Transm. Distrib., Vol. 143, No. 5, Septembev 1996.

5. Satish J Ranade : ‘Classical Analysis First-swing transient stability ,

Lecture 3’, EE532 Power System Dynamics and Transients.

6. Narain G. Hingorani , Understanding FACTS


Documents

report on first swing stability