The concept of the stability of the parallel operation of power systems. Dynamic stability of power systems

DYNAMIC STABILITY
ENERGOSYSTEM

If
static
steadiness
characterizes
which has established the operating mode of the system, then at
dynamic stability analysis reveals
the ability of a system to maintain a synchronous mode
work under large disturbances. Large
disturbances arise for various short
short circuits, disconnections of power lines,
generators, transformers, etc. To large
disturbances also include power changes
large load, loss of excitation of any
generator, inclusion of large engines. One
from the consequences of the disturbance that has arisen is
deviation of the rotational speeds of the rotors of the generators
from synchronous - oscillation of the rotors of generators.

DYNAMIC STABILITY OF POWER SYSTEMS

If, after some perturbation, the mutual angles of the vectors
will take certain values ​​(their oscillations will decay around
any new values), then it is assumed that the dynamic
stability is maintained. If at least one generator
the rotor starts to turn relative to the stator field, then
this is a sign of a violation of dynamic stability. All in all
the case of the dynamic stability of the system can be judged by
dependences f t obtained as a result of joint
solution of the system of equations of motion of the rotors of generators. But
there is a simpler and more intuitive method based on
energy approach to the analysis of dynamic stability,
which is called graphical method or method
areas.

Consider the case where the power plant is running
through the double-circuit line to the tires endless
power (Figure 14.1, a). Constancy condition
system bus voltage (U const) excludes
rocking of the rotors of the generators of the receiving system and
much
simplifies
analysis
dynamic
sustainability. The system equivalent circuit is shown
in Figure 14.1, b. The generator is included in the equivalent circuit
transition resistance X d and EMF Eq.

Analysis of the dynamic stability of the simplest system by the graphical method

Analysis of the dynamic stability of the simplest system by the graphical method

The power supplied by the generator to the system is
is equal to the turbine power and is denoted by P0
, injection
generator rotor - 0. Power characteristic,
the corresponding
normal
(pre-emergency)
regime, we write without taking into account the second harmonic that
quite
permissible
v
practical calculations.
Taking Eq E, we obtain the expression for the characteristic
power in the following form:
E U
P
sin
X d
where
, (14.1)
X d X d X T 1 X L1 // X L 2 X T 2.
The dependence for the normal mode is shown in
Figure 14.1, d (curve 1).

Analysis of the dynamic stability of the simplest system by the graphical method

Suppose the L2 line is suddenly disconnected.
Consider the operation of the generator after it is turned off.
Equivalent circuit of the system after shutdown
shown in Fig. 14.1, c. Total resistance
post-emergency mode X d (p.a) X d X T 1 X L1 X T 2
will increase
on
comparison
with X d (total
resistance of normal mode). This will cause
reduction of the maximum power characteristic
post-emergency mode (curve 2, Fig. 14.1, d).
After a sudden disconnection of the line,
transition
with
specifications
power
1
on
characteristic 2. Due to the inertia of the rotor, the angle is not
can change instantly, so the operating point
moves from point a to point b.

Analysis of the dynamic stability of the simplest system by the graphical method

On the shaft connecting the turbine and generator,
there is an excess torque equal to the difference
turbine power, which did not change after
disconnection of the line, and new power of the generator
P P0 P (0). Under the influence of this difference, the rotor
the car begins to accelerate, moving in
side of large angles
... This movement
superimposed on the rotation of the rotor with a synchronous
speed, and the resulting rotation speed
rotor will be equal to 0, where 0 is synchronous
rotational speed; - relative speed.

Analysis of the dynamic stability of the simplest system by the graphical method

As a result of the acceleration of the rotor, the operating point
moves along characteristic 2. Power
generator increases, and the excess (accelerating)
moment (proportional to the difference P P0 P (0)) -
decreases. The relative speed increases to
points with. At the point with excess torque becomes
equal to zero, and the speed is maximum.
The rotation of the rotor at speed does not stop at
point c, the rotor by inertia passes this point and
continues to move. But the excess torque at
this changes its sign and starts to brake the rotor.

Analysis of the dynamic stability of the simplest system by the graphical method

The relative velocity decreases and at point d
becomes zero.
The angle at this point reaches its maximum
values. But at point d the relative motion
the rotor does not stop, because on the rotor shaft
the generator has a braking excess torque,
therefore
rotor
starts
traffic
v
the opposite side, i.e. towards point c.
The rotor passes point c by inertia, near point b
the angle becomes minimal and a new one begins
the cycle of relative movement of the rotor. Attenuation
vibrations of the rotor due to energy losses at
relative movement of the rotor.

Analysis of the dynamic stability of the simplest system by the graphical method

Excess torque is associated with excess power
expression
M
where
R
,
- the resulting rotor speed.
Swing speed change is negligible
is small compared to speed 0, so with
an error sufficient for practice can be
take 0, and then we get (expressing M, P and 0
in relative units) M * R
0
0 1 .
, insofar as

Analysis of the dynamic stability of the simplest system by the graphical method

Considering
only
relative
the movement of the rotor and the work done when
this movement, when the rotor moves on
infinitesimal angle d redundant
the moment does elementary work
M d. With no loss, all work
goes to change the kinetic energy
the rotor in its relative motion.

Analysis of the dynamic stability of the simplest system by the graphical method

During the period of movement when excess
moment
accelerates
rotation
rotor,
kinetic energy stored by the rotor in
the period of its acceleration will be determined by
formula
0
Fusk Рd f abc
0
,
where f abc is the shaded area abc on
fig. 11.1, d.

Analysis of the dynamic stability of the simplest system by the graphical method

Change in kinetic energy
braking is calculated as
rotor
v
his
m
Ftorm Рd f cde
0
.
Squares f abc
and
f cde proportional to
kinetic energy of acceleration and deceleration,
are called areas of acceleration and deceleration.
During the braking period, the kinetic energy
the rotor transforms into potential energy, which
increases with decreasing speed.

Analysis of the dynamic stability of the simplest system by the graphical method

At point d, the kinetic energy is zero, and for
determining the maximum angle of deflection of the rotor
it is enough to fulfill the condition
max
Fusk Ftorm
,
thus, at the maximum deflection angle
the acceleration area is equal to the braking area.
Maximum possible braking area
determined by the angle cr. If the maximum angle
exceeds the value of cr, then on the rotor shaft of the generator
an accelerating excess torque (P0 PG) will appear and
the generator will be out of sync.

Analysis of the dynamic stability of the simplest system by the graphical method

In Fig. 14.1, d, the area cdm is the maximum
possible area of ​​acceleration. Having defined
it, you can estimate the stock of dynamic
sustainability.
Coefficient
stock
is determined by the formula
Fcdm Fabc
Kz
100%
Fabc
.

The most common type of disturbance in which
an analysis of the dynamic stability in the system is required,
is a short circuit. Consider the general case
asymmetrical short circuit at the beginning of the line to
Figure 14.2, a. The equivalent circuit of the system for the short circuit mode is shown
(n)
in Figure 14.2, b. Additional reactance X included in
point of short circuit, depends on the type of short circuit, and is determined
same as item 2 .: X (2) X 2, X (1) X 2 X, 0 X (1,1) X 2 // X 0, where X 2
and X 0 - total resistances of reverse and zero
sequences respectively. After the occurrence of a short circuit
the power transferred from the generator to the system will change,
as well as the total resistance of the direct sequence,
connecting the generator to the system.

Dynamic stability at short circuits in the system

Dynamic stability at short circuits in the system

At the moment of a short circuit due to a change in the parameters of the circuit
there is a transition from one characteristic
power to another (Figure 14.3). Since the rotor
possesses
mechanical
inertia,
then
injection
and the given
the generator power is reduced to the value P (0).
At the same time, the turbine power does not change in view of
delays of its regulators. On the rotor of the generator
appears
some
excess
moment,
determined by the excess power (P P0 P (0)). Under
by the action of this moment, the rotor of the generator starts
accelerate, the angle increases.

Dynamic stability at short circuits in the system

Qualitatively, the process proceeds in the same way as in
the previous case of a sudden disconnection of the line.
Since the L2 line, like any other element
power system, has protection, through a certain
time it will be disconnected by switches B1 and B2. it
time is calculated as
toff tsz toff
,
where tsz
- actual time of protection operation;
toff - operation time of switches B1 and B2.

Dynamic stability at short circuits in the system

The time toff corresponds to the angle of tripping of the short circuit off.
Disconnection of the short circuit causes a transition from the characteristic
emergency mode power 2 per characteristic
post-emergency mode 3. In this case, the sign changes
excess
moment;
he
turns
from
accelerating to decelerating. The rotor, braking,
continues to move in the direction of increasing the angle due to the kinetic
energy. This movement will continue until those
until the braking area f dcfg equals
area of ​​acceleration f abcd.

Dynamic stability at short circuits in the system

But the movement of the rotor does not stop, since on it
acts
brake
excess
moment,
determined by excess power Рtorm Р f Р0. Rotor,
accelerating, begins to move in the opposite direction.
Its speed is maximum at point n. After point n
the relative speed starts to decrease and
becomes zero at point z. This point
is determined from the equality of the areas f nefgd and f xnz.
Due to energy losses, the rotor oscillations will
decay around a new equilibrium position
post-emergency mode - point n.

Dynamic stability at short circuits in the system

With a three-phase short circuit at the beginning of the line
mutual
resistance
schemes
becomes
infinitely large, since the resistance
reactance X (3) 0. In this case, the power characteristic
emergency mode coincides with the abscissa
(Figure 14.4).
Rotor
generator
starts
his
relative motion under the influence of excess
moment equal to the mechanical moment of the turbine.
The differential equation of the rotor motion at
this has the form
Tj
d 2
dt
2
P0
.
(14.4)

Analysis of a three-phase short circuit by a graphical method

This equation is linear
analytical solution. Let's rewrite
(14.4) in the following form
d P0
2
dt T j
dt
and has
the equation
d 2
,
whence taking the integral of the left and right sides,
get
P0
t c1
Tj
.
(14.5)

Analysis of a three-phase short circuit by a graphical method

At t 0, the relative speed of the rotor is 0 and,
therefore, c1 0. Integrating again
(14.5), we get
P0 t 2
c2
Tj 2
.
The constant of integration c2 is determined from
conditions: 0, c2 0 at t 0. Finally, the dependence
angle from time has the form
2
P0 t
0
Tj 2
.(14.6)

Analysis of a three-phase short circuit by a graphical method

The limiting angle of tripping of a three-phase short circuit can
be determined from expression (14.3), simplified
condition Pmax 2 0:
cos off
P0 cr 0 Pmax 3 cos cr
Pmax 3
.

Analysis of a three-phase short circuit by a graphical method

Limit trip time at three-phase short circuit
is determined from expression (14.7):
toff pr
2T j off pr 0
P0
.

The equation of motion of the rotor is nonlinear and cannot
be decided analytically. The exception is
full power reset in emergency mode, i.e.
Rabbi max 0 discussed above. The equation
(14.4)
solved
methods
numerical
integration. One of them is the method
consecutive intervals, illustrating
physical picture of the process.
According to this method, the whole rocking process
the rotor of the generator is divided into a number of intervals
time t and for each of them sequentially
the increment of the angle is calculated.

Solution of the equation of motion of the rotor by the method of successive intervals

At the moment of short-circuit, the power supplied by the generator
falls and there is a certain excess of power P (0).
For a small time interval t, we can assume that
that the excess power during this interval
remains unchanged. Integrating expression (14.4),
at the end we get at the end of the first interval
d
t 2
V (1) (0) t c1, (1) (0)
c2.
dt
2

Solution of the equation of motion of the rotor by the method of successive intervals

The relative speed of the rotor at the moment of short circuit is equal to
zero (c1 0), and therefore the relative velocity
rotor at the end of the first interval is equal to V (1). At
t 0 angle 0, therefore c2 0. Acceleration 0 can
be calculated from (9.1):
0
P (0)
Tj
,
this implies
(1)
P (0) t 2
Tj 2
.

Solution of the equation of motion of the rotor by the method of successive intervals

Here angle and time are expressed in radians. V
in practical calculations, the angle is expressed in degrees, and
time - in seconds:
(hail)
t (c)
360 f
0
t (rad)
(0)
(glad)
, (14.8)
. (14.9)

Solution of the equation of motion of the rotor by the method of successive intervals

Using (14.8) and (14.9) and taking into account that
T j (c)
T j (rad)
0
,
we get
(1)
P (0)
360 f t P (0)
0
0 K
Tj
2
2
2
,
where
360 f t 2
K
Tj
.
(14.10)

Solution of the equation of motion of the rotor by the method of successive intervals

The acceleration generated in the second interval is
proportional to the excess power at the end of the first
interval P (1). When calculating the increment of the angle in
during the second interval, it is necessary to take into account that
that in addition to the acceleration acting in this interval
(1) the rotor already has a speed V (1) at the beginning of the interval:
(2) V (1) t
where
(1) t 2
2
V (1) t K
P (1)
, (14.11)
2
P (1) P0 Pmax sin 1
.

Acceleration (0)
changes during the first
interval
time,
therefore
for
decline
errors in calculating the speed value V1
it is necessary to assume that on the first interval
medium acceleration is in effect
(0) Wed
(0) (1)
2
.

Then the relative
formula
speed
V (1) (0) av t
(0) (1)
2
will
expressed
t.
Substituting this expression into (14.11), we obtain
(2)
or
(0) (1)
2
t
2 (1) t 2
2
(0) t 2
2
(2) (1) K R (1)
(1) t 2,
.

Angle increment on subsequent
calculated in the same way:
intervals
(n) (n 1) K P (n 1).
If at the beginning of some K - interval occurs
short circuit shutdown, then the excess power suddenly
varies from some value P (K 1) (Figure 14.6)
P (K 1)
before
, which corresponds to the transition from
characteristics 1 to 2.

To determine the excess power when switching from one mode (1)
to another (2)

Angle increment at first
short circuit tripping is defined as
(K) (K 1) K
interval
after
P (K 1) P (K 1)
2
. (14.12)
Calculation by the method of successive intervals
held until the angle
won't start
decrease, or it becomes clear that the angle
grows indefinitely, i.e. machine stability
is violated.

Payment
dynamic
sustainability
complex
is executed in the following sequence.
systems
1. Calculation of the normal operation of the electrical system
before the occurrence of a short circuit. The calculation results in the values
EMF of power plants (Ei) and the angles between them.
2. Drawing up equivalent circuits of reverse and zero
sequences and determination of their resulting
resistances relative to the point of short circuit and point of zero
potential of the circuit. Calculation of additional reactances
X (n) corresponding to the considered SC.
3. Calculation of own and mutual conductances for all
power plants of the system in emergency and post-emergency
modes.

Dynamic stability of complex systems

4. Calculation of angular displacements of machine rotors using
method of successive intervals. Definition of values
the power delivered by the machines at the beginning of the first interval:
P1 E12Y11 sin 11 E1E2Y12 sin 12 12 ...
P2 E2 E1Y21 sin 21 21 E22Y22 sin 22 ...
…………………………………………………..
5. Definition
interval:
excess
P1 (0) P10 P1
P2 (0) P20 P2
power
v
the beginning
the first
,
,
………………….
where P, P
etc. - the power generated by machines in
20
10
the moment preceding the short circuit.

Dynamic stability of complex systems

6. Calculation of the angular displacements of the rotors of generators in
during the first interval t:
1 (1) K1
2 (1) K 2
P1 (0)
2
P2 (0)
,
,
2
……………………
In the second and subsequent intervals, the expressions for the angular
displacements have the form:
1 (n) 1 (n 1) К1 Р1 (n 1)
,
2 (n) 2 (n 1) K 2 P2 (n 1)
,
………………………………..
Coefficients K are calculated in accordance with the expression
(14.10).

Dynamic stability of complex systems

7. Determination of the values ​​of the angles at the end of the first -
the beginning of the second intervals
1 (n) 1 (n 1) 1 (n)
,
2 (n) 2 (n 1) 2 (n)
,
…………………………
where 1 (n 1), 2 (n 1), etc. - angle values ​​at the end
previous interval.

Dynamic stability of complex systems

8. Finding new values ​​of mutual angles
rotor divergence:
12 1 2
,
13 1 3
,
…………….
Having determined these values, proceed to the calculation
next interval, i.e. calculates the power in
the beginning of this interval, and then the calculation is repeated,
starting from item 5.

Dynamic stability of complex systems

At the moment of disabling the damage, all own
and the mutual conductances of the branches change. Corner
displacement of the rotors in the first time interval
after disconnection are counted for each
machines by expression (14.12).
Calculation of the dynamic stability of complex systems
performed
for
a certain
time
short-circuit tripping and continues not only until the moment
short circuit shutdown, and until there is
the fact of violation of stability or its
conservation. This is judged by the nature of the change
relative angles.

Dynamic stability of complex systems

If at least one corner grows indefinitely
(for example, angle 12 in Figure 14.7), then the system is considered
dynamically unstable. If all mutual angles
tend to fade near any
new values, the system is stable.
If the nature of the change in relative angles
a violation of the stability of the system was established when
taken at the beginning of the calculation of the short circuit trip time,
then to determine the limiting short-circuit time it follows
repeat the calculation, reducing the short-circuit disconnection time to
until stable work is ensured
power systems.

Dynamic stability is understood as the ability of the power system to maintain synchronous parallel operation of generators in the event of significant sudden disturbances in the power system (short circuit, emergency shutdown of generators, transformer line).

The area method is used to assess the dynamic stability. As an example, consider the operating mode of a double-circuit power transmission connecting the power plant with the power system, with a short circuit on one of the lines with disconnection of the damaged line and its successful automatic reclosure (Figure 10.3, a).

The original power transmission mode is characterized by point 1 located on the angular characteristic I, which corresponds to the original power transmission scheme (Figure 10.3, b).

Rice. 10.3. Qualitative analysis of dynamic stability at K3 on the power transmission line: a - power transmission scheme; b - angular characteristics of power transmission; в - change of angle in time

At K3 at point K1 on line W2, the angular characteristic of the power transmission takes position II. The decrease in the amplitude of characteristic II is caused by a significant increase in the resultant resistance between the points of application. At the moment K3, the electric power is reset by an amount due to a decrease in the voltage on the station buses (point 2 in Fig. 10.3, b). The discharge of electrical power depends on the type of K3 and its location. In the extreme case, with three-phase K3 on the buses of the station, the power is reset to zero. Under the influence of the excess mechanical power of the turbines over the electrical power, the rotors of the station generators begin to accelerate, and the angle increases. The process of changing the power goes according to characteristic II. Point 3 corresponds to the moment of disconnection of the damaged line on both sides by relay protection devices RZ. After disconnecting the line, the transmission mode is characterized by point 4 located on the characteristic, which corresponds to the transmission scheme with one disconnected line. During the change in the angle from to, the rotors of the station generators acquire additional kinetic energy. This energy is proportional to the area bounded by the line, characteristic II and ordinates at points 1 and 3. This area is called the acceleration area. At point 4, the process of braking the rotors begins, since the electric power is greater than the power of the turbines. But the braking process occurs with increasing angle. The increase in the angle will continue until all the stored kinetic energy is converted into potential.

The potential energy is proportional to the area bounded by the line and the angular characteristics of the post-fault mode. This area is called the braking area. At point 5, after a certain pause after disconnecting the W2 line, the automatic reclosure device is triggered (it is assumed to use a three-phase high-speed automatic reclosure with a short pause). If the automatic reclosure is successful, the process of increasing the angle will continue according to the characteristic (point 6), corresponding to the original power transmission scheme. The increase in the angle will stop at point 7, which is characterized by equal areas. At point 7, the transient process does not stop: due to the fact that the electric power exceeds the power of the turbines, the braking process will continue according to the characteristic, but only with a decrease in the angle. The process will establish itself at point 1 after several fluctuations around this point. The nature of the change in angle 5 in time is shown in Fig. 10.3, c.

In order to simplify the analysis, the power of the turbines during the transient process is assumed unchanged. In fact, it changes somewhat due to the action of the turbine speed controllers.

Thus, the analysis showed that under the conditions of this example, the stability of parallel operation remains. A necessary condition for dynamic stability is the fulfillment of the conditions for static stability in the post-accident mode. In the considered example, this condition is met, since the power of the turbines does not exceed the static stability limit.

The stability of parallel operation would be broken if in the transient process the angle went over the value corresponding to point 8. Point 8 limits the maximum braking area to the right. The angle corresponding to point 8 is called critical. When this boundary is crossed, an avalanche increase in the angle is observed; loss of generators from synchronism.

The dynamic stability margin is estimated by a coefficient equal to the ratio of the maximum possible braking area to the acceleration area:

When the mode is stable, when there is a violation of stability.

In case of unsuccessful autoreclosing (switching on the line to the not removed K3), the process from point 5 will go to characteristic II. It is easy to make sure that under the conditions of this example, stability after repeated K3 and subsequent disconnection of the line is not preserved.

The steady-state operating mode of the power system is quasi-steady, since it is characterized by small changes in active and reactive power flows, voltage and frequency values. Thus, in the power system, one steady-state operating mode is constantly shifting to another steady-state operating mode. Small changes in the operating mode of the power system occur as a result of an increase or decrease in the consumption of consumer electrical installations. Small perturbations cause the system to respond in the form of oscillations in the rotational speed of the rotors of the generators, which can be increasing or damping, oscillatory or aperiodic. The nature of the resulting vibrations determines the static stability of this system. Static stability is checked during prospective and detailed design, development of special automatic control devices (calculations and experiments), commissioning of new system elements, changes in operating conditions (integration of systems, commissioning of new power plants, intermediate substations, power lines).

The concept of static stability is understood as the ability of the power system to restore the original or close to the original mode of operation of the power system after a small disturbance or slow changes in the parameters of the mode.

Static stability is a prerequisite for the existence of a steady state of operation of the system, but does not predetermine the ability of the system to continue operating in the event of finite disturbances, for example, short circuits, switching on or off of power lines.

There are two types of violations of static stability: aperiodic (sliding) and oscillatory (self-swinging).

Static aperiodic (sliding) stability is associated with a change in the balance of active power in the power system (change in the difference between electrical and mechanical power), which leads to an increase in the angle δ, as a result, the machine may fall out of synchronicity (violation of stability). The angle δ changes without fluctuations (aperiodic), at first slowly, and then ever faster, as if sliding (see Fig. 1, a).

Static periodic (oscillatory) stability is associated with the settings of the automatic excitation regulators (ARV) of the generators. ARV should be configured in such a way as to exclude the possibility of self-swinging of the system in a wide range of operating modes. However, with some combinations of repairs (circuit-mode situation) and settings of excitation regulators, fluctuations in the control system may occur, causing increasing fluctuations in the angle δ up to the machine falling out of synchronism. This phenomenon is called self-swinging (see Fig. 1, b).

Fig. 1. The nature of the change in the angle δ in violation of static stability in the form of sliding (a) and self-swinging (b)

Static aperiodic (creep) stability

The first stage in the study of static stability is the study of static aperiodic stability. In the study of static aperiodic stability, it is assumed that the probability of oscillatory disturbance of stability with an increase in the flow through intersystem connections is very small and self-swinging can be neglected. To determine the area of ​​aperiodic stability of the power system, the operating mode of the power system is made heavier. The weighting method consists in sequentially changing the parameters of nodes or branches, or their groups at specified steps, followed by calculating a new steady state at each step of change and is performed as long as the calculation is possible.

Consider the simplest network diagram, which consists of a generator, power transformer, power line, and infinite power buses (see Fig. 2).

Fig. 2. Computational circuit equivalent circuit

In the simplest case under consideration, the electromagnetic power that can be transmitted from the generator to the tires of infinite power is described by the following expression:

In the written expression, the variable is the line voltage module on the station buses, reduced to the HV side, and the variable is the line voltage module at the point of the buses of infinite power.

Fig. 3. Vector voltage diagram

The mutual angle between the voltage vector and the voltage vector is denoted through the variable -, for which the counterclockwise direction from the voltage vector is taken as the positive direction.

It should be noted that the formula for the electromagnetic power is written on the assumption that the generator is equipped with an automatic excitation regulator, which controls the voltage on the side of the generator voltage (), and also for simplicity of calculations, the active resistance in the elements of the design circuit was neglected.

Analyzing the formula for electromagnetic power, we can conclude that the amount of power transmitted to the power system depends on the angle between the voltages. This dependence is called the angular characteristic of the power transmission (see Fig. 4).

Fig. 4. Angular power characteristic

The steady-state (synchronous) mode of operation of the generator is determined by the equality of two moments acting on the turbine generator shaft (we believe that the moment of resistance due to friction in the bearings and the resistance of the cooling medium can be neglected): turbine torque Mt, rotating the rotor of the generator and striving to accelerate its rotation, and the synchronous electromagnetic moment Ma'am, counteracting the rotation of the rotor.

Suppose that steam enters the turbine of the generator, which creates a torque on the turbine shaft (at some approximation, it is equal to the external torque Mvn transmitted from the prime mover). The steady-state operating mode of the generator can be at two points: A and B, since at these points a balance is observed between the turbine torque and the electromagnetic torque, taking into account losses.

point A an increase / decrease in the turbine power by ΔP will lead to an increase / decrease in the angle d, respectively. Thus, the equilibrium of the moments acting on the rotor shaft is maintained (equality of the turbine torque and the electromagnetic torque, taking into account losses), and thus a violation of the synchronous machine with the network does not occur.

When a synchronous machine is running in point V increasing / decreasing the turbine power by ΔP will lead to a decrease / increase in the angle d, respectively. Thus, the balance of the moments acting on the rotor shaft is disturbed. As a result, either the generator falls out of synchronicity (i.e., the rotor starts to rotate at a frequency that differs from the frequency of rotation of the stator magnetic field), or the synchronous machine moves to a point of stable operation (point A).

Thus, from the considered example, it can be seen that the simplest criterion for maintaining static stability is the positive sign of the expression, which determines the ratio of the power increment to the angle increment:

Thus, the area of ​​stable operation is determined by the range of angles from 0 to 90 degrees, and in the range of angles from 90 to 180 degrees, stable parallel operation is impossible.

The maximum value of power that can be transferred to the power system is called the static stability limit, and corresponds to the power value at a mutual angle of 90 degrees:

Operation at the maximum power corresponding to an angle of 90 degrees is not performed, since small disturbances that are always present in the power system (for example, load fluctuations) can cause a transition to an unstable region and a violation of synchronism. The maximum allowable value of the transmitted power is taken to be less than the static stability limit by the value of the static aperiodic stability safety factor for active power.

Static stability margin for power transmission in normal mode must be at least 20%. The value of the permissible active power flow in the controlled section according to this criterion is determined by the formula:

The margin of static stability for power transmission in the post-emergency mode must be at least 8%. The value of the permissible active power flow in the controlled section according to this criterion is determined by the formula:

Static periodic (oscillatory) stability

An incorrectly selected control law or incorrect setting of the parameters of the automatic excitation regulator (ARV) can lead to a violation of oscillatory stability. In this case, the violation of oscillatory stability can occur in modes not exceeding the limiting mode in terms of aperiodic stability, which has been repeatedly observed in operating electric power systems.

The study of oscillatory static stability is reduced to the following stages:

1. Drawing up a system of differential equations that describes the considered electric power system.

2. The choice of independent variables and the linearization of the written equations in order to form a system of linear equations.

3. Compilation of the characteristic equation and determination of the area of ​​static stability in the space of adjustable (independent) settings of the ARV.

The stability of a nonlinear system is judged by the attenuation of the transient process, which is determined by the roots of the characteristic equation of the system. To ensure stability, it is necessary and sufficient that the roots of the characteristic equation have negative real parts.

To assess the stability, various methods of analyzing the characteristic equation are used:

1. algebraic methods (Routh method, Hurwitz method) based on the analysis of the coefficients of the characteristic equation.

2. frequency methods (Mikhailov, Nyquist, D-partitioning) based on the analysis of frequency characteristics.

Measures to increase the static stability limit

Measures to increase the static stability limit are determined by analyzing the formula for determining the electromagnetic power (the formula is written under the assumption that the generator is equipped with an automatic excitation regulator):

1. Application of ARV strong action on generating equipment.

One of the effective means of increasing static stability is the use of powerful ARV generators. When using ARV devices of generators of strong action, the angular characteristic is modified: the maximum of the characteristic is shifted to the range of angles greater than 90 ° (taking into account the relative angle of the generator).

2. Maintaining voltage at network points using reactive power compensation devices.

Installation of reactive power compensation devices (SK, CShR, STK, etc.) to maintain voltage at network points (lateral compensation devices). The devices allow you to maintain voltages at network points, which has a beneficial effect on the static stability limit.

3. Installation of longitudinal compensation devices (UPC).

With an increase in the length of the line, its reactance accordingly increases and, as a result, the limit of the transmitted power is significantly limited (the stability of parallel operation deteriorates). Reducing the reactance of a long transmission line increases its transmission capacity. To reduce the inductive resistance of the power transmission line, a longitudinal compensation device (LCC) is installed in the line cut, which is a battery of static capacitors. Thus, the resulting line resistance is reduced, thereby increasing the throughput.

1.1. The concept of static and dynamic stability in electric power systems

The stability of the state of an electrical system is understood as its ability to restore the original mode (or close enough to it) after the impact of any disturbance ("large" or "small"). The process of breaking stability in electrical systems is always associated with the limited capacity of its individual elements - communication lines, transformers, etc. Naturally, with unchanged parameters of the electrical system, the limit of the transmitted power depends on the voltage levels and losses of the transmitted power at the resistances of the elements. Violations of stability in electrical systems occur as a result of the impact on its operation of disturbing factors, which can be "large" and "small". The course of the process is the same in this case and is accompanied in any case by a sharp decrease in voltage in the nodes of the system (the appearance of an "avalanche" of voltage), an increase in the current in its branches, and a change in the speed of rotation of electrical machines. Impaired stability always ends with the appearance of an asynchronous stroke associated with an unlimited change in the rotation speed of synchronous machines, and often leads to the "collapse" of the system - disconnecting the load, station generators, and dividing the system into asynchronous parts. "Small" disturbances are dangerous for the operation of electrical systems in severe modes, when power flows close to the limiting ones flow through its elements. Whereas "large" perturbations can cause a violation of stability in normal modes. Depending on the cause that led to the violation of stability, three types are distinguished: - static stability - the ability of the system to maintain (restore) the original (or close to it) regime under the action of "small" disturbances. - dynamic stability - the ability of the system to restore a long-term steady-state regime under "large" disturbances. - net resilience - the ability of the system to return to a long-standing steady state after a short-term violation of stability.

Static stability of a synchronous generator

The assessment of the static stability of a synchronous generator connected to the buses of the power system (Fig. 1) can be performed using Newton's second law for a rotating body

where M in - the torque on the shaft of the power engine, kgm; M s - moment of resistance (braking moment) on the generator shaft, kg.m; ω - angular frequency of rotation of the unit shaft, s -1;

Moment of inertia, kg.m.s 2; GD 2 - flywheel masses of rotating parts connected to the shafts of the power engine and generator, kg m 2; g = 9.81 m / s 2 - acceleration of gravity.

1. Scheme of power transmission from a synchronous generator to the power system and its equivalent circuit: T - turbine; Г - generator; T1 - substation transformer; L1, L2 - power lines; T2 - transformer for communication with the power system; ES - power system.

The static stability of a synchronous unit is estimated at a constant synchronous speed, at which the power on the shaft of the power engine and the synchronous generator is proportional to the moments, and in relative units are equal, i.e.

Static stability is assessed with the relative movement of the rotor of the unit, i.e., when the rotor moves relative to the vector of the rotating electromagnetic field of the generator stator (Fig. 2), when the angle of the rotor departure is changed. The rate of its change corresponds to the derivative (1.1.2)

With the relative motion of the generator rotor, the equation of motion (1.1.1) can be represented in the following form:

(1.1.3)

Rice. 2. Basic design diagrams of synchronous generators: a - implicit-pole; b - salient

This equation is the equation of dynamic equilibrium, because with equality R T = P r the angle of departure of the rotor 0 has a constant value. If there is no equality of powers, then either the acceleration of the unit takes place at P T > P G , or deceleration at R T < Р d, that is, by the sign of the power difference, one can judge the nature of the movement of the unit shaft. Therefore, it is advisable to use equation (1.1.3) in the following form

(1.1.4)

where ∆Р- excess power. Power engine power characteristic in coordinates R, is a straight line, since the power developed by the engine does not depend on the rotor angle.

Power characteristic of a synchronous generator in coordinates R, is represented by a sinusoidal angular characteristic (Fig. 3) obtained from a vector diagram:

for an implicit pole machine (turbine generator)

(1.1.5)

for salient pole machine (hydrogenerator)

(1.1.6)

where resistances of generators in the longitudinal and transverse axes, taking into account the resistances of the equivalent circuit (see Fig. 1)

Pa fig. 3 shows the characteristics of the turbine and generator. The characteristics have two points of mutual intersection 1 and 2. In accordance with the position of theoretical mechanics at points

STATIC STABILITY

power system - ability electric power system restore the initial state (mode) after small disturbances. S.'s violation at. can occur when transmitting large powers through power lines (as a rule, extended ones), with a decrease in voltage at load nodes due to a shortage of reactive power, when generators of power plants are operating in under-excitation mode. Main measures of ensuring S. at .: increase in the nominal value. power transmission line voltage and reduction of their inductive resistance; automatic excitation control large synchronous machines, application synchronous compensators, synchronous motors and static. reactive power compensators at load nodes. S. at. can be increased also when used in power systems of generators with excitation control in the longitudinal and transverse rotor windings.

Big Encyclopedic Polytechnic Dictionary. 2004 .

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• Aircraft aerodynamics. Part two. Equilibrium in straight flight and static stability, Pyshnov V.S. This book will be produced according to your order using Print-on-Demand technology. Aircraft aerodynamics. Part two. Balance in straight flight and static stability ...