Dynamic stability- the ability of the system to return to its original state after a large disturbance. Maximum size- a solution in which a very small increase in load causes a violation of its stability. Element Bandwidth systems are called the highest power, cat. can be transmitted through the element, taking into account all limiting factors. Positional system- such a system, in a cat. Parameters of parameters depend on the current state, the relative position, regardless of how this state was achieved. At the same time, the real dynamic characteristics of the electric system. are replaced by static ones. Static characteristics- these are connections between the parameters of the system, presented analytically or graphically and independent of time. Dynamic characteristics– connections of pairs obtained under the condition that they depend on time. Voltage reserve: k u =. Power reserve: k R =. Assumptions made in stability analysis: 1. Rotation speed of synchronizing machine rotors during electromechanical flow. The PP varies within small limits (2-3%) of the synchronous speed. 2. The voltage and currents of the stator and rotor of the generator change instantly. 3. The nonlinearity of system pairs is usually not taken into account. The nonlinearity of the pairs of r-ma is taken into account; when such consideration is refused, this is stipulated and the system is called linearized. 4.Move from one district of the electrical system. to others, it is possible by changing the own and mutual resistance circuits, the EMF of generators and motors. 5. The study of dynamic stability under asymmetrical disturbances is carried out in a direct sequence scheme. The movement of the rotors of generators and engines is caused by the moments created by direct sequence currents. Problems of dynamic stability analysis associated with the transition of the system from one steady state to another. A) calculation of dynamic pairs transition during operational or emergency shutdown of loaded elements of the electrical system. b) determination of dynamic pairs transitions during a short circuit in the system, taking into account: - possible transition of 1 asymmetrical short circuit to another; - work of automatic restarting of an electrical device that has switched off after a short circuit. The results of the calculation of dynamic stability are: - the maximum time for disconnecting the calculated type of short circuit at the most dangerous points of the system; - system pauses Automatic reclosure installed on various elements of the electrical system; - par-ry syst. automatic transfer of reserve (ATR).
The electric power system is dynamically stable, if, under any strong disturbance, the synchronous operation of all its elements is maintained. To clarify the fundamental provisions of dynamic stability, let us consider the phenomena that occur when one of the two parallel power line circuits is suddenly disconnected (Fig. A). The resulting resistance in normal mode is given by the expression 
,
and after disconnecting one of the circuits - by the expression
Since , then the relation is valid 
If one of the power transmission line circuits suddenly turns off, the rotor does not have time to instantly change the angle δ due to inertia. Therefore, the mode will be characterized by the point b on another angular characteristic of the generator - characteristic 2
in Fig. 
After reducing its power, an excess accelerating torque arises, under the influence of which the angular velocity of the rotor and the angle δ increase. As the angle increases, the generator power increases according to the characteristic 2 . During the acceleration process, the generator rotor passes 61.1. point With, after which its torque becomes leading. The rotor begins to slow down and, starting from the point d its angular velocity decreases. If the angular velocity of the rotor increases to a value = point e, then the generator falls out of synchronization. The stability of the system can be judged by the change in angle δ over time. The change in δ shown in Fig. A, corresponds to stable operation of the system. When δ changes along the curve shown in Fig. b, the system is unstable.
distinctive features of static and dynamic stability: at static stability during the appearance of disturbances, the power of the generator changes according to the same angular characteristic, and after their disappearance, the system parameters remain the same as before the appearance of disturbances; for dynamic installation it is the other way around.
Analysis of the dynamic stability of the simplest systems by graphical method. If static stability characterizes the steady state of the system, then the analysis of dynamic stability will reveal the ability of the system to maintain a synchronous operating mode under large disturbances. Large disturbances occur during various short circuits, disconnection of power lines, generators, transformers, etc. One of the consequences of the resulting disturbance is the deviation of the rotation speeds of the generator rotors from synchronous. If, after some disturbance, the mutual angles of the rotors take certain values (their oscillations die out around some new values), then it is considered that dynamic stability is maintained. If at least one generator's rotor begins to rotate relative to the stator field, then this is a sign of a violation of dynamic stability. In the general case, the dynamic stability of the system can be judged from the dependences b = f
(t), obtained as a result of the joint solution of the equations of motion of generator rotors. Analysis of the dynamic stability of the simplest system by graphical method. Let's consider the simplest case, when a power plant G operates through a double-circuit line to buses of infinite power (see Fig. a). a - schematic diagram; b - equivalent circuit in normal mode; c - equivalent circuit in post-emergency mode; d - graphic illustration of dynamic transition: characteristics of normal and emergency modes (curves 1, 2, respectively). The condition of constant voltage on the system buses ( U
=
const) eliminates swinging of the receiving system generators and greatly simplifies the analysis of dynamic stability. The power characteristic corresponding to normal (pre-emergency) mode can be obtained from the expression 
without taking into account the second harmonic, which is quite acceptable in practical calculations. Taking E q =
E, then .
Let's assume that the line L 2 suddenly turns off. Let's consider the operation of the generator after it is turned off. The replacement circuit of the system after disconnecting the line is shown in Fig.,c. The total resistance of the post-emergency mode will increase compared to X dZ(total resistance of normal mode). This will cause a decrease in the maximum power characteristic of the post-emergency mode (curve 2, Fig. d). After a sudden shutdown 61.2.
line there is a transition from power characteristic 1 to characteristic 2. Due to the inertia of the rotor, the angle cannot change instantly, so the operating point moves from the point A to point b. An excess torque appears on the shaft, determined by the difference between the turbine power and the new generator power (P = P 0 - P (0)). Under the influence of this difference, the rotor machine begins to accelerate, moving towards larger angles. This movement is superimposed on the rotation of the rotor at synchronous speed, and the resulting rotor speed will be w = w 0 + , where w 0 is the synchronous speed of rotation; - relative speed. As a result of the acceleration of the rotor, the operating point begins to move along characteristic 2. The generator power increases, and the excess torque decreases. The relative speed increases to a point With. At the point With the excess torque becomes zero and the speed becomes maximum. The movement of the rotor with speed does not stop at the point With, the rotor by inertia passes this point and continues to move. But the excess torque changes sign and begins to slow down the rotor. The relative rotation speed begins to decrease at the point d becomes equal to zero. The angle at this point reaches its maximum value. But also at the point d the relative movement of the rotor does not stop, since an excess braking torque acts on the shaft of the unit, so the rotor begins to move towards the point With, the relative speed becomes negative. Full stop With the rotor passes by inertia, near the point b the angle becomes minimal and a new cycle of relative motion begins. Angle fluctuations (t) are shown in Fig., d. The damping of oscillations is explained by energy losses during relative motion of the rotor. Excess torque is associated with excess power by the expression
,
where ω is the resulting rotor rotation speed. 
The steady-state operating mode of the power system is quasi-steady, as it is characterized by small changes in the flow of active and reactive power, voltage values and frequency. Thus, in the power system, one steady state of operation constantly transitions to another steady state of operation. Small changes in the operating mode of the power system arise due to an increase or decrease in the consumption of consumer electrical installations. Small disturbances cause a system reaction in the form of oscillations in the speed of rotation of the generator rotors, which can be increasing or decreasing, oscillatory or aperiodic. The nature of the resulting vibrations determines the static stability of a given system. Static stability is checked during long-term and detailed design, development of special automatic control devices (calculations and experiments), commissioning of new system elements, changes in operating conditions (consolidation 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 mode parameters.
Static stability is a necessary condition the existence of a steady-state operating mode of the system, but does not predetermine the ability of the system to continue operation when finite disturbances occur, for example, short circuits, switching on or off power lines.
There are two types of violations of static stability: aperiodic (sliding) and oscillatory (self-swinging).
Static aperiodic (creeping) stability is associated with a change in the balance of active power in the power system (a change in the difference between electrical and mechanical powers), which leads to an increase in the angle δ, as a result of which the machine may fall out of synchronism (stability violation). The angle δ changes without oscillations (aperiodically), first slowly, and then faster and faster, as if sliding (see Fig. 1, a).
Static periodic (oscillatory) stability is associated with the settings of automatic excitation regulators (AEC) of generators. The AVR must be configured in such a way as to exclude the possibility of self-oscillation of the system in a wide range of operating modes. However, with certain combinations of repairs (circuit-mode situation) and settings of the excitation regulators, oscillations in the control system may occur, causing increasing fluctuations in the angle δ until the machine falls out of synchronism. This phenomenon is called self-swinging (see Fig. 1, b).
Fig.1. The nature of the change in angle δ when static stability is violated in the form of sliding (a) and self-swinging (b)
Static aperiodic (sliding) stability
The first stage of the study of static stability is the study of static aperiodic stability. When studying static aperiodic stability, it is assumed that the probability of an oscillatory violation 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 of sequentially changing the parameters of nodes or branches, or their groups in specified steps, followed by the calculation of a new steady state at each step of change and is carried out as long as the possibility of calculation is ensured.
Let's consider the simplest scheme network, which consists of a generator, power transformer, power line and infinite power buses (see Fig. 2).
Fig.2. Equivalent circuit of calculation circuit
In the simplest case under consideration, the electromagnetic power that can be transferred from the generator to the infinite power buses is described by the following expression:
In the written expression, the variable represents the module of the linear voltage at the station buses, reduced to the HV side, and the variable is the module of the linear voltage 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 by 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 electromagnetic power is written on the assumption that the generator is equipped with an automatic excitation regulator that controls the voltage on the generator voltage side (), and 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 power transmission power (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 caused by friction in the bearings and the resistance of the cooling medium can be neglected): turbine moment Mt, which rotates the generator rotor and tends to accelerate its rotation, and the synchronous electromagnetic torque Ma'am, counteracting the rotation of the rotor.
Let us assume that steam enters the generator turbine, which creates a torque on the turbine shaft (to 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 maintained between the turbine torque and the electromagnetic torque, taking into account losses.
point A An increase/decrease in turbine power by an amount ΔP will lead to an increase/decrease in angle d, respectively. Thus, the balance 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 the disruption of the synchronous machine with the network does not occur.
When a synchronous machine operates in point IN An increase/decrease in turbine power by an amount ΔP will lead to a decrease/increase in angle d, respectively. Thus, the balance of moments acting on the rotor shaft is disrupted. As a result, either the generator falls out of synchronism (i.e. the rotor begins to rotate at a frequency different from the rotation speed magnetic field stator), or the synchronous machine moves to the point of stable operation (point A).
Thus, from the considered example it is clear that the simplest criterion for maintaining static stability is the positive sign of the expression that determines the ratio of the power increment to the angle increment:
Thus, the region of stable operation is determined by the range of angles from 0 to 90 degrees, and in the region 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 carried out, 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 permissible value of 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.
The static stability margin for power transmission in normal mode should be at least 20%. The value of the permissible flow of active power in the controlled section according to this criterion is determined by the formula:
The static stability margin for power transmission in post-emergency mode must be at least 8%. The value of the permissible flow of active power in the controlled section according to this criterion is determined by the formula:
Static periodic (oscillatory) stability
An incorrectly selected control law or incorrect settings of the parameters of the automatic excitation controller (AEC) can lead to a violation of oscillatory stability. In this case, a violation of oscillatory stability can occur in modes not exceeding the limiting mode for aperiodic stability, which has been repeatedly observed in existing electric power systems.
The study of oscillatory static stability comes down to the following steps:
1. Drawing up a system of differential equations that describes the electrical power system under consideration.
2. Selecting independent variables and performing linearization of the written equations in order to form a system of linear equations.
3. Drawing up a characteristic equation and determining the region of static stability in the space of adjustable (independent) ARV settings.
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 stability, use various methods analysis of the characteristic equation:
1. algebraic methods (Rouse method, Hurwitz method), based on the analysis of the coefficients of the characteristic equation.
2. frequency methods (Mikhailov, Nyquist, D-partition method), 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 electromagnetic power (the formula is written under the assumption that the generator is equipped with an automatic excitation regulator):
1. Use of strong action ARVs on generating equipment.
One of effective means increasing static stability is the use of strong action ARV generators. When using strong-action ARV generator devices, the angular characteristic is modified: the maximum of the characteristic shifts 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, UShR, STC, etc.) to maintain voltage at network points (transverse 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 (LPD).
As the length of the line increases, its reactance increases accordingly and, as a result, the limit of transmitted power is significantly limited (the stability of parallel operation deteriorates). Reducing the reactance of a long power line increases its capacity. To reduce the inductive reactance of a power transmission line, a longitudinal compensation device (LPD), which is a battery of static capacitors, is installed in the line section. Thus, the resulting line resistance decreases, thereby increasing the throughput.
Physical foundations of stability of electric power systems Static stability of a power system is stability under small disturbances of the regime. From consideration of the simplest mechanical systems it follows that there are states (regimes) in which the system, after a random disturbance, tends to restore the original or close to it mode. In other regimes, a random disturbance takes the system away from its initial state. In the first case, the system is stable, in the second – unstable.
Physical foundations of stability of electric power systems In steady state, there is a balance between the source energy entering the system and the energy expended in the load and to cover losses. With any disturbance, manifested in a change in the mode parameter, this balance is disrupted. If the system has such properties that energy after a disturbance is consumed more intensively than it is produced by power plants, then the new regime that arose as a result of the disturbance cannot be provided with energy and the previous steady state or one close to it must be restored in the system. Such a system is stable.
Physical foundations of stability of electric power systems From the definition of stability it follows that the condition for maintaining the stability of the system (stability criterion) is a ratio, or in differential form. The quantity is called excess energy. This energy is positive if the additional generated energy that appears during the disturbance increases more intensely than the load of the system taking into account losses in it.
Physical foundations of stability of electric power systems Under this condition, the stability criterion will be written in the form, i.e., the mode is stable if the derivative of the excess energy with respect to the defining parameter is negative.
Physical foundations of stability of electric power systems To ensure the stability of the system, the margin of its static stability, which is characterized by the shift angles of the generator rotors and voltage vectors at the nodal points of the system, is essential. Great importance has a reserve of static stability in post-emergency mode - in terms of electrical transmission power it should be 5 - 10%, in normal mode 15 - 20%. However, these numbers are not strictly limited.
Physical foundations of stability of electric power systems To check the static stability of the system, it is necessary to draw up differential equations small vibrations for all its elements and control devices, and then explore the roots of the characteristic equation for stability. Since a rigorous solution to such a problem is very difficult, engineering calculations use approximate methods for studying stability, which are based on the use of practical stability criteria.
Static stability of the system “equivalent generator - constant voltage buses” A system in which a single remote power plant is connected to constant voltage buses (system) is called the simplest (Fig. 11. 1, a). It is believed that the total power of the electrical stations of the system significantly exceeds the power of the station in question. This allows us to consider the voltage on the system buses unchanged under any operating modes. The simplest system is also called a single-machine model of a power system or a “machine-bus” model.
STATIC STABILITY The analyzed power plant is connected through transformer connections and a power transmission line to the generators of a powerful concentrated power system, so powerful that its receiving buses are designated as infinite power buses (IBP). Distinctive features The BBM is a voltage that is constant in modulus and a constant frequency of this voltage. When using BBMs, the corresponding power systems in electrical diagrams, as a rule, are not depicted. In equivalent circuits, infinite power buses are used as an element representing a powerful system.
STATIC STABILITY In Fig. 11. 1, b shows two main thermal units power station: turbine and generator. The turbine torque depends on the amount of supplied energy: for steam turbine- this is steam, for a hydraulic turbine - water. In normal mode, the main parameters of the energy carrier are stable, so the torque is constant. The power supplied by the generator to the system is determined by several parameters, the influence of which depends on the power characteristics of the generator.
STATIC STABILITY To obtain the generator power characteristics, a vector diagram of power transmission was constructed (Fig. 11. 1, c). Here full vector the current is decomposed into its real and imaginary components, and the resistance is obtained from the equivalent circuit of the system shown in Fig. 11. 1, g:
STATIC STABILITY From the vector diagram it follows that, where is the active component of the current, is the shift angle of the EMF vector relative to the voltage vector. Multiplying both sides of the equality by, we obtain, (11. 1) where is the active power supplied by the generator (taken in relative units).
STATIC STABILITY Dependence (11.1) is sinusoidal in nature and is called the generator power characteristic. At constant generator EMF and voltage, the angle of rotation of the generator rotor is determined only by its active power, which in turn is determined by the power of the turbine. The power of the turbine depends on the amount of energy carrier, and in coordinates it is represented by a straight line.
STATIC STABILITY At certain values of the emf of the generator and the voltage of the receiving side, the power characteristic has a maximum, which is calculated by the formula. (11.2) The value is also called the “ideal” power limit of the electrical system. Each turbine power value corresponds to two intersection points of characteristics a and b (Fig. 11.2, a), at which the generator and turbine powers are equal.
STATIC STABILITY Let us consider the operating mode at point a. If the generator power is increased by an amount, then the angle, following a sinusoidal dependence, will change by an amount. From Fig. 11.2, but it follows that at point a a positive increment in power corresponds to a positive increment in angle. When the generator power changes, the balance of the turbine and generator torques is disrupted. As the power of the generator increases, a braking torque appears on the rotor shaft connecting to the turbine, exceeding the torque of the turbine. The braking torque causes the generator rotor to slow down, which causes the rotor and the associated EMF vector to move towards a decreasing angle (Fig. 11. 2, b).
STATIC STABILITY It must be emphasized that the movement of the rotor under the influence of excess torque is superimposed on its movement in the positive direction with a synchronous speed that is many times higher than the speed of this movement. As a result, at point a the original operating mode is restored and, as follows from the definition of static stability, this mode is stable. The same conclusion can be obtained by reducing the generator power at point a.
STATIC STABILITY If you reduce the power of the generator at point b, then an accelerating excess torque appears on the generator rotor shaft, which increases the angle. As the angle increases, the power of the generator decreases further, this leads to an additional increase in the accelerating torque, thus, an avalanche-like process occurs, which is called loss of synchronism. The process of falling out of synchronism and the asynchronous mode in which the generator ultimately finds itself is characterized by a continuous movement of the EMF vector relative to the voltage of the receiving system.
STATIC STABILITY If the generator power is increased at point b, then an excess braking torque will arise, which will cause the operating point of the turbine-generator system to move to point a. Thus, point a of the power characteristic is a point of stable equilibrium of the moments of the turbine and generator, point b is a point of unstable equilibrium. Similarly, all points lying on the increasing part of the power characteristic are points of stable operation of the system, and points lying on the falling part of the characteristic are points of unstable operation. The boundary between the zones of stable and unstable operation is the maximum power characteristic.
STATIC STABILITY Thus, a sign of static stability of an electrical system is the sign of the power increment relative to the angle increment. If, then the system is stable; if this ratio is negative, then it is unstable. Passing to the limit, we obtain the stability criterion the simplest system: . An increase in turbine power from value to (Fig. 11. 2, a) leads to an increase in the rotor angle from value to value and to a decrease in static stability.
STATIC STABILITY Obviously, under operating conditions, the generator should not be loaded to its maximum power, since any slight deviation in the mode parameters can lead to loss of synchronism and the generator switching to asynchronous mode. In the event of unforeseen disturbances, a reserve for generator loading is provided, characterized by a static stability safety factor. (11.3)
STATIC STABILITY Guidelines for the stability of power systems stipulate that in normal modes of power systems, a stability margin of the power transmission connecting the station with the buses of the power system must be ensured of at least 20% in normal mode and 8% in short-term post-emergency mode. In the most severe modes, in which an increase in power flows along the lines makes it possible to reduce consumer restrictions or losses of hydraulic resources, the stability margin can be reduced to 8%. By short-term we mean post-emergency conditions lasting up to 40 minutes, during which the dispatcher must restore the normal static stability margin.
Characteristics of the power of a salient-pole generator To characterize the power of a salient-pole machine, we will write down the expression for the active power supplied to the system. Considering that we will rewrite in the form, the expression for power
Power characteristic of a salient-pole generator From the last expression it follows that the power characteristic of a salient-pole generator, in addition to the main sinusoidal component, contains a second component - the second harmonic component, the amplitude of which is proportional to the difference in inductive reactances and. The second harmonic shifts the maximum power characteristic towards a decreasing angle (Fig. 11. 3). The first, main part depends on the magnitude of the EMF, which indicates that the generator must be excited. The second component does not depend on the excitation of the generator; it shows that a salient-pole generator can produce active power without excitation due to the reactive torque, but this active power depends on the sine of the double angle.
Power characteristic of a salient-pole generator The amplitude of the power characteristic increases compared to the characteristic of a non-salient-pole machine. But this increase appears only at low EMF values (when the first and second components are of the same order). Under normal conditions, the amplitude of the second harmonic is 10–15% of the main harmonic and does not affect significant influence on the power characteristic.
Power characteristics of a generator with ARV Let's assume that the generator in Fig. 11. 1 voltage regulation system is disabled. Let's construct a vector diagram of the system under consideration, highlighting in it the voltage on the generator buses (Fig. 11. 4, a). It depends on the voltage drop across the external resistance of the system: where is the system. external resistance
Characteristics of generator power with ARV The voltage vector on the generator buses divides the voltage drop vector into two parts, proportional to the inductive reactances and. Let us increase the transmitted active power by and thereby the angle by. This will cause a change in the reactive power transferred to the system. To obtain the dependence of reactive power on the angle, we write the expression following from the vector diagram shown in Fig. 11. 1, at
Characteristics of the power of a generator with ARV Multiplying the left and right sides of the last equality by, we obtain. Having expressed, from the last relationship, we obtain an expression for the reactive power supplied by the generator from the angle: .
Characteristics of the power of a generator with ARV From the diagram it follows that an increase in the angle causes a decrease in the voltage on the generator buses. Let's assume that the automatic excitation regulator is turned on and controls the voltage. When this voltage decreases, the regulator increases the excitation current, and with it the EMF, until the previous voltage value is restored. When considering the steady-state operating conditions of a generator with ARV at various angle values, one often assumes a constant voltage. In Fig. 11. 4, b shows a family of characteristics constructed for different meanings EMF.
Characteristics of the power of a generator with ARV If we take point a as the starting point of the normal mode, then to increase the power (accompanied by an increase in the angle), the points of the new steady-state modes will be determined by the transition from one characteristic to another in accordance with the vector diagram (Fig. 11. 4, a) . By connecting the points established at different excitation levels, we obtain the external characteristic of the generator. It increases even in
Characteristics of generator power with ARV Proportional-type regulators (RPT) with gain factors of 50... 100 make it possible to maintain the voltage on the generator busbars almost constant. The gain is defined as the ratio of the numbers of excitation units and generator voltage units. But the maximum transmission power of such a generator, equipped with an ARV with such a gain, is slightly higher than the maximum power of an unregulated generator.
Characteristics of the power of a generator with ARV This is due to the fact that with an increase in power, at a certain point in the power characteristics (point 3 in Fig. 11.5, a), self-oscillation of the generator begins, i.e. periodic oscillations of the rotor with increasing amplitude lead to the generator falling out of synchronicity. Therefore, proportional-type regulators do not try to maintain it, allowing it to slightly decrease with increasing load. In this case, the maximum power that can be achieved is significantly higher than the power (Fig. 11.5, b).
Power characteristic of a generator with ARV The power characteristic at gain factors of the order of 20... 40 has approximately the same maximum as the generator characteristic at. Consequently, a generator equipped with a proportional type regulator can be represented in equivalent circuits by transient EMF and resistance.
Power characteristic of a generator with ARV The power characteristic of a generator replaced by an EMF can be obtained in the same way as the characteristic of a salient-pole generator
Characteristics of the power of a generator with ARV If the RPT has a dead zone, the mode at o is considered critical, i.e. the maximum power is reached at point at
Characteristics of the power of a generator with ARV The regulator begins to work only after the voltage deviation in one direction or another reaches a certain value. For smaller deviations lying in the dead zone, the regulator does not work. The boundaries of the dead zone correspond to two external characteristics (Fig. 11. 6).
Characteristics of the power of a generator with ARV Let point a correspond to the initial mode. With a slight disturbance that causes an increase in the angle, the voltage on the generator busbars decreases, but the regulator does not work as long as the angle deviation lies in the dead zone. As the angle increases, an accelerating excess torque appears on the generator shaft, causing it to further increase. When the angle of motion crosses the dead zone boundary (point b), the controller begins to operate.
Power characteristic of a generator with ARV Increasing the excitation current, and, consequently, the EMF of the generator, slows down the decrease in power, moving the operating point on the power characteristic, corresponding to large EMF (points c, d). At point e the excess power becomes zero, but due to the inertia of the rotor the angle continues to increase. At point f the angle becomes maximum, after which it begins to decrease.
Characteristics of the power of a generator with ARV After point g has been passed, lying on external characteristics, the regulator will begin to reduce the exciter voltage and the power change curve will cross the internal power characteristics in the opposite direction. Thus, due to internal instability, undamped oscillations of the generator rotor (angle oscillations) occur. The amplitude of these oscillations depends on the width of the controller dead zone. The voltage, power and current of the generator fluctuate along with the angle. Such fluctuations make it difficult to control the operation of the generator and make it necessary to abandon its operation in such modes.
Characteristics of the power of a generator with ARV o. It is possible to ensure stable operation of the generator by using more complex excitation regulators that respond not only to changes in voltage, but also to the speed and even acceleration of changes in voltage. Such regulators are called strong regulators. Strong-action regulators provide a constant voltage at the terminals of the generator (without self-oscillation), so a generator equipped with such a regulator can be represented by a constant voltage source with zero resistance when calculating static stability in an equivalent circuit.
STATIC STABILITY
electrical power system - ability electric power system restore the original state (regime) after small disturbances. Violation of S. can occur when transmitting large powers through power lines (usually long ones), when the voltage in load nodes decreases due to a shortage of reactive power, when power plant generators operate in underexcitation mode. Basic measures to ensure S. u.: increase in nominal. power line voltages and reduction of their inductive reactance; automatic excitation control large synchronous machines, application synchronous compensators, synchronous electric motors and static reactive power compensators in load nodes. S. u. can also be increased when using generators in power systems with excitation control in the longitudinal and transverse windings of the rotor.
Big Encyclopedic Polytechnic Dictionary. 2004 .
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English: Energetic system static (resistance) stability The ability of a power system to return to a steady state after minor disturbances (according to GOST 21027 75) Source: Terms and definitions in the electric power industry. Directory… Construction dictionary
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- , V. Pyshnov. Aerodynamics of the aircraft. Part two. Equilibrium in straight flight and static stability Reproduced in the original author's spelling of the 1935 edition (ONTI publishing house...
- Aerodynamics of the aircraft. Part two. Equilibrium in straight flight and static stability, Pyshnov V.S. This book will be produced in accordance with your order using Print-on-Demand technology. Aerodynamics of the aircraft. Part two. Balance in straight flight and static stability...
The stability of an aircraft is its ability to maintain a given balancing flight mode without intervention and return to it after the cessation of external disturbances. Stability is conventionally divided into static and dynamic. An aircraft is statically stable if, with a small change in the angles of attack, slip and roll, forces and moments arise aimed at restoring the original flight mode. Dynamic stability is characterized by the damping of transient processes of perturbed motion.
The controllability of a rocket is its ability to perform, in response to purposeful actions of the pilot, any maneuver provided for during operation under acceptable flight conditions. Balancing flight modes are modes in which the forces and moments acting on the rocket are balanced, and the static controllability of the rocket is characterized by the deflections of the controls, movements of the control levers and forces on them required for balancing the rocket.
There are concepts of longitudinal and lateral static stability. Longitudinal static stability is understood as the property of a rocket, after the cessation of external disturbances, to return without pilot intervention to the initial values of the angle of attack and flight speed, and by lateral stability - to the initial values of the roll and slide angles. Accordingly, controllability characteristics are usually divided into longitudinal and lateral.
To achieve the goal, it is necessary to complete a number of tasks:
· Analyze the concept of aircraft stability;
· Describe static stability and ways to ensure it;
The flight of an aircraft occurs under the influence of aerodynamic force, engine thrust and gravity. To ensure flight and perform the flight mission, the rocket must adequately respond to control influences - targeted changes in aerodynamic force and thrust force, i.e. be controlled.
Small, previously unknown deviations (perturbations) of the aerodynamic force and thrust force from the calculated values, not related to control, also change the motion of the aircraft. To perform a flight, the rocket must withstand these disturbances, i.e. be resilient.
Stability and controllability are important properties, determining the possibility of flight along a given trajectory. When studying stability and controllability, the aircraft is considered as a material body and its motion is described by the equations of motion of the center of mass and rotation around the center of mass. The movement of the center of mass and its rotation relative to the center of mass are related. However, the joint study of these movements is very difficult due to large number equations describing general motion.
In real motion, as a rule, the following conditions are met: firstly, the deflection of the controls almost instantly leads to a change in the aerodynamic forces acting on the rocket, and secondly, the resulting control forces are significantly less than the main aerodynamic forces.
These conditions allow us to assume that the angular motion, in contrast to the movement of its center of mass, can be changed quite quickly and, therefore, the movement (rotation) relative to the center of mass and the movement of the center of mass along the trajectory can be considered separately.
During flight, in addition to the main ones, the rocket is subject to small disturbing forces associated with wind and turbulent atmospheric disturbances, changes in the rocket configuration, thrust pulsation and other reasons. Therefore, the real motion of the rocket is perturbed and differs from the unperturbed one. The disturbing forces are unknown in advance and are random in nature, so in the equations of motion it is almost impossible to accurately specify all the forces acting on the rocket in flight.
Stability is the property of a rocket to restore the kinematic parameters of undisturbed motion and return to the original mode after the cessation of the influence of disturbances on the rocket.
When performing individual stages of flight, it is necessary to be able to purposefully influence the nature of the rocket’s movement, that is, control the rocket.
When controlling a rocket, the following tasks are solved:
· providing the required values of kinematic parameters necessary for the implementation of a given reference motion;
· parrying disturbing influences and maintaining specified or close to them motion parameters under the influence of disturbance.
These problems can be solved if the rocket reacts properly, responds to control inputs, that is, it has controllability.
Controllability is the property of responding with appropriate linear and angular movements in space to the deflection of controls
There is a conventional division of rocket motion stability into static and dynamic. The static stability of a rocket characterizes the balance of forces and moments in the reference steady motion. A rocket is called statically stable with respect to one or another parameter of motion, in which the deviation of this parameter from the reference value immediately after the cessation of the disturbance leads to the appearance of a force (in forward movement) or moment (in angular) aimed at reducing this deviation. If forces and moments are aimed at increasing the initial deflection, then the rocket is statically unstable.
Static stability is an important factor in assessing the dynamic stability of a rocket, but does not guarantee it, since when determining dynamic stability, it is not the initial tendency to eliminate the disturbance that is assessed, but the final state - the presence of asymptotic stability or instability in the sense of A.M. Lyapunova. When assessing dynamic stability, it is important not only the final state (stable or unstable), but also indicators of the process of attenuation of deviations from unperturbed motion:
· decay time of deviations in motion parameters;
· nature of the perturbed motion (oscillatory, aperiodic);
· maximum deviation values;
· period (frequency) of oscillations (if the process is oscillatory), etc.
The distance between the center of gravity and the neutral center point is called the aircraft's static stability margin.
In order to be more precise in statements about rocket stability, it is necessary to introduce two aspects of this topic that have not previously been mentioned. First, the effect of the initial disturbance mainly depends on whether the control surfaces deflect or not during the subsequent motion. It is obvious that two extreme possibilities must be assumed, namely, the controls are permanently in their original position and they are completely free to move on their hinges. The first assumption corresponds very closely to the example of a rocket with power-driven control surfaces, which are usually irreversible in the sense that aerodynamic forces cannot cause them to deflect against the control mechanism. The second limiting case - the controls are free - is a somewhat idealized representation of a rocket with manual control mode, where the pilot allows the rocket to fly in "automatic mode". The degree of sustainability of these extreme examples can vary, so much so that, obviously, the desired sustainability goals under both permanent and free controls can sometimes be very difficult to achieve.
The second side of the stability problem, which has not been previously considered, is the influence of the propulsion system. Stability with both the engine running and the engine not running needs to be considered. The difference arises mainly due to two factors: one of them is the direct effect of thrust on the balance and movement of the rocket; the second is the change in aerodynamic forces acting on the wing and tail due to the flow caused by the propulsion system. The latter factor is generally more significant in rockets driven by propellers than in rockets powered by jet engines; it is called the influence of the co-current jet from propeller. Even in jet rockets, most designers place the tail surfaces quite high above the jet stream to avoid mutual harmful effects.
Bibliography
1. Balakin, V.L., Lazarev, Yu.N. Aircraft flight dynamics. Stability and controllability of longitudinal movement. – Samara, 2011.
2. Bogoslovsky S.V. Dorofeev A.D. Flight dynamics of aircraft. – St. Petersburg: GUAP, 2002.
3. Efimov V.V. Aviation Basics. Part I. Fundamentals of aerodynamics and flight dynamics of aircraft: Tutorial. – M.: MSTU GA, 2003.
4. Karman, T. Aerodynamics. Featured topics in their historical development. – Izhevsk: Research Center “Regular and Chaotic Dynamics”, 2001
5. Starikov Yu.N., Kovrizhnykh E.N. Fundamentals of aircraft aerodynamics: Textbook. allowance. –2nd ed., revised. and additional – Ulyanovsk: UVAU GA, 2010.
