Static stability of electric power systems.

Static stability is the ability of a system to restore its original or close to its original state after its disturbance.

Dynamic stability is the ability of a system to restore its original or close to its original state after a large disturbance.

Based on the definition static stability of the system, it can be concluded that there is such a mode in which a very small increase in loads causes a violation of its stability. This mode is called limiting, and the system loads are called maximum or ultimate loads according to the conditions of static stability.

The electric power system must operate in such a way that some changes (deterioration) in the regime do not lead to a violation of the stability of its operation. The simplest estimate of its stability margin is based on a comparison of the indicators of the tested (initial) mode and the indicators characterizing the regime that is limiting in terms of stability.

Static stability of EPS operation in post-emergency modes is ensured, as a rule, through measures that do not require additional capital investments:

- short-term increase in voltage at the terminals of generators;

– rapid decline power transmission loads by disconnecting some of the generators at power plants, etc.

- In addition, there are measures that increase static stability, but require some capital investment:

- the use of a high-speed generator excitation system;

- use of synchronous compensators at intermediate substations;

- use of static thyristor compensators;

- longitudinal capacitive compensation of inductive resistance of power transmission using static capacitors, etc.

- Practically all these measures allow to increase the dynamic stability.

In operation, in cases where it is necessary to prevent the restriction of consumers or the loss of water resources, long-term operation of power transmission in normal mode with a margin of static stability reduced to 5-10% is allowed, depending on the role of power transmission in the power system and the consequences of a possible violation of stability.

The exact answer to the question of the stability (or instability) of the system can be obtained by calculating all the roots of the characteristic equation. However, the procedure for calculating the roots for the equations high order belongs to the category of extremely labor-consuming, therefore, a number of special mathematical conditions have been developed that allow, without calculating the roots of the characteristic equation, to determine their location on the complex plane and thus accurately answer the question of the stability or instability of the system. These mathematical conditions are called stability criteria. Distinguish between algebraic and frequency stability criteria. Algebraic tests contain a group of conditions (a group of inequalities), composed by certain rules from the coefficients of the characteristic equation, under which stability takes place. If at least one of them is violated, then instability takes place. To carry out an analysis using algebraic criteria, it is obviously necessary to pre-calculate the coefficients of the polynomial on the left side of the characteristic equation. Necessary and sufficient conditions for the stability of a linear homogeneous system of differential equations in the form of algebraic inequalities were established by the English scientist Rouse and the Swiss mathematician Hurwitz.

Algebraic stability criteria:

o Hurwitz criterion

The system of Hurwitz inequalities is constructed as follows. The coefficients of the characteristic polynomial are used to compose the Hurwitz square matrix. Necessary and sufficient conditions for stability are that all n diagonal minors must be positive.

o Routh criterion

It is more convenient for high-order systems with numerically specified coefficients of the characteristic equation. From the coefficients of the characteristic polynomial, a Routh table is compiled, each element of which is calculated through four elements of the two preceding rows. The calculation algorithm is clearly visible from the table. In total, the table contains (n ​​+ 1) rows. The Routh stability requirements are formulated as follows: for the system to be stable, it is necessary and sufficient that all the coefficients of the first column be positive.

Frequency stability criteria.

In the practice of studying the stability of systems, there are cases when it is difficult not only to calculate the roots of the characteristic equation, but also to obtain the equation itself in the form of a characteristic polynomial on the left-hand side. In such cases

more convenient are the frequency criteria, which,

as well as algebraic criteria, it is possible to determine the presence or absence of roots of the characteristic equation in the right half-plane on the root plane. Frequency criteria are based on the principle of the argument known in higher mathematics. ...

PART 2

STABILITY OF ELECTRICAL SYSTEMS

Chapter 9

STATIC STABILITY

9.1. BASIC CONCEPTS AND DEFINITIONS OF STABILITY

Division of electrical system modes into steady-state and transient conditional. In the steady state of a real systemI its parameters are constantly changing, which is associated with the following factors:

- change in load and response to these changes in controloperating devices;

- normal operational changes to the comsystem mutations;

- switching on and off individual generators or from changing their power.

Thus, in the steady state of the system, there is alwayssmall perturbations of the parameters of its regime, at which it shouldnot to be stable.

Static stability - this is the ability of the system toto establish the initial (or close to the initial) regime after its small disturbance.

Emergency modes in the electrical system occur whenShort circuit, emergency shutdowns of loaded units or lines andetc. Under the influence of large perturbations, abrupt changes in the regime occur.

Dynamic stability - it is the ability of the system to return to its original (or close to it) state after painyour indignation. When, after a great indignation, the synchronousthe system mode is violated, and then after an allowable break is restored, then they talk about the resulting sustainability systems. The resulting resilience is sometimes considered differentlyvisibility of dynamic stability, sharing the synchronousdynamic stability and the resultingdynamic stability.

Based on the definition of the static stability of the systemit can be concluded that there is a mode in which a very small increase in loads causes a violation of its stability.This mode is called limiting, and the system loads -maximum or ultimate loads according to conditionspits of static stability.

The limitation of loads can also be caused by other circumstances, for example, heating of the elements of the electrical system (generators, transformers, etc.). In this case, they talk aboutlimit loads according to heating conditions and set alsothe maximum lifetime of the regime.

Limitations of loads by voltage levels in knots are possiblelakh, crown tension, etc.

Bandwidth element of the system is called themore power that can be transmitted through this element withtaking into account all limiting factors (heating, stability,stress in nodes, etc.). Sometimes the bandwidth is determineddivide by one factor and speak, for example, of the heating capacity.

The concept of bandwidth is also true for dinmicrostability. In this case, they talk about the limit transmitted power under the conditions of dynamicstability in case of short-circuit at any point, disconnection of the line, etc.The problems arising in the analysis of stability are very complexus and voluminous. Therefore, to understand physical entity racesof the phenomena under consideration, they resort to simplifying the problems to be solved. Sometimes it is necessary to abandon the mathematical rigor of the solution, to discard secondary factors. This does not reflect the details, but a fairly complete picture of the phenomenon is obtained. One trick to simplify the solution is to view the electrical system as positional.

Positional system - such a system in which the parametersregime depend on current state, the relative position of, for example, rotors of generators and motors, regardless of how this state was achieved. In this case, the real dynamic characteristics of the system elements are replaced by static ones.

Static characteristics - these are mode parameters linkssystems presented analytically or graphically and are not dependentfrom time to time. These connections are revealed mainly by establishingthe current mode of the system.

Dynamic characteristics are the connections of parameters, semigiven that they depend on time. In this casereflects the influence of the first, and possibly higher productionwater parameters under consideration.

To describe the positional system, static characteristics are sufficient. Dynamic characteristics allow you to study the electrical system as dynamic.

The dynamic transition from one mode to another is subject to qualitative assessment... At the same time, the nature of the protetransient process (fast, slow, monotonous,aperiodic) and the nature of the new steady-state regime. Scithe quality of the transient is good if observedits fast decay, aperiodicity or monotony are given. The post-transient regime must havea sufficient margin of stability, which is checked fromby changing a parameter. The largest deviation at which the system is still stable is determined bythe stability margin, expressed by the safety factor. For example, the voltage margin is calculated by the formula

power reserve - according to the formula

The new steady state can be estimated usingquality criteria established by GOST.

9.2. ASSUMPTIONS ACCEPTED IN THE ANALYSIS OF STABILITY

In addition to the assumptions made in the analysis of electromagnetic transient processes, several more assumptions are made,simplifying the assessment of sustainability and ensuring sufficientaccuracy for engineering calculations.

1. It is assumed that the rotational speed of synchronous rotorsmachines during the flow of electromechanical transient processes varies within small limits (2 ... 3%) synchronous speed.

2. It is believed that the voltage and currents of the stator and rotor of the generatorthe torus changes instantly.

3. The nonlinearity of the system parameters is usually not taken into account.The nonlinearity of the mode parameters, on the contrary, is taken into account. Kogyes, they refuse such accounting, they specifically stipulate this,the system is called linearized.

4. Change from one electrical system mode to anotherit is possible by changing the intrinsic and mutual resistances of the circuit, andSee also EMF of generators and motors.

5. Investigation of dynamic stability with asymmetry perturbations are performed in a direct sequenceness. It is believed that the movement of the rotors of generators andmotors is due to the moments created by the currents of the direct
sequence.

9.3. STABILITY CALCULATION PROBLEMS ELECTRICAL SYSTEMS

When analyzing static stability, a number of problems arise,which are solved in the design and operating organizations.These tasks include:

1. Calculation of the parameters of limiting modes (limitingpower supplied through the lines of the power system, criticalvoltages of the nodal points of the system supplying the load, etc.).

2. Determination of the values ​​of the safety factors. Together with atintroduced in Sec. 9.1 voltage safety factors andpower, safety factors can be calculated by tuningparameters of ARV:

font-size: 11.5pt; color: black; letter-spacing: -.4pt "> where TO max and

Kmin - maximum and minimum value tuningparameters corresponding to the boundary of the region of the static sustainability.

3. Selection of measures to improve static stabilitypower systems or providing a given throughput transmission.

4. Development of requirements aimed at improving the stabilitysystem flexibility. The ARV setting is selected that provides the required voltage maintenance accuracy.

The solution of the listed tasks is carried out taking into account the possibility of self-swinging of the system.

The problems of dynamic stability analysis are related to the transitionhome system from one steady state to another. itthe following tasks:

a) calculation of dynamic transition parameters during operationon-line or emergency shutdown of loaded elementselectrical system.

b) determination of the parameters of dynamic transitions atshort circuits in the system, taking into account various factors:

- possible transition of one asymmetric short circuit to another(for example, single phase to two phase);

Works of automatic reclosing of the element, fromswitched on after a short circuit, etc. The results of calculating the dynamic stability are:- the maximum shutdown time of the calculated type of short circuit in the mostdangerous points of the system;

- pauses of automatic reclosing systems installed on various elementselectrical system;

- parameters of automatic transfer switch systems (ATS).
Calculations are carried out, as a rule, taking into account nonlinearities and

significant dynamic characteristics.

9.4. STATIC STABILITY OF THE SIMPLE SYSTEM

The simplest system is understood as one in which a single power plant (equivalent generator) is connected to aby us (system) constant voltage transformers and lines that transmit power from the station to the system(fig.9.1, a). It is assumed that the total power of electricalstations of the system are many times higher than the power of the consideredinstalled station. This allows you to read the voltage on the buses of the system.themes unchanged ( U = const ) under any modes of its operation.

In fig. 9.1, b two main heating units are presentedpower plant: turbine and generator. The turbine rotor is driven into rotation by steam supplied to the turbine from the electric boiler.reeds. Turbine torque depends on the quantity

Energy carrier. For a steam turbine, this is steam, for hydrotouringbin - water. In normal operation, the main steam meters of energy carrier - temperature and steam pressure are stable,therefore, the turbine torque is constant. Power, yougiven by the generator to the system is determined by several steammeters, the influence of which depends on the power characteristic generator.

font-size: 9.0pt; color: black; letter-spacing: -.05pt "> Figure 9.1. Assessment of static stability the simplest system: a - principleschematic diagram of the system; b - turbine - generator unit; v - generator vector diagram; G - system equivalent circuit; d - mechanical analogue of the block

turbine - generator

To obtain the power characteristic, a vectorpower transmission diagram (Fig.9.1, v). It repeats the diagram shown in fig. 2.10, a however, the total current vector in it isreplaced by its real and imaginary components, and the resistance decay xd - on resistancexdΣ schema-derived substituteof the system shown in Fig. 9.1, G:

xdΣ.= Xd + xT1 + xL2 / xL2 + xT2

It follows from the vector diagram that

Ia xdΣ = E sinδ,

where Iа ​​is the active component of the current; δ - EMF shear angle E fromwith respect to voltageU. Multiplying both sides of the equality byU/ xdΣ, get

(9.1)

where R - active power supplied by the generator (adopted inrelative units).

Dependence (9.1) has a sinusoidal character and callsthe characteristic of the generator power. At constantEMF E generator and voltageU generator swing angleis determined only by its active power, which, in itsturn is determined by the power of the turbine. A clear illustration of the dependence of the power (torque) of the turbine on the shear angle 8is a system of two discs connected by springs (Fig.9.1,e). In XX mode (without regard to friction) driving (rotor field, connectedturbine) and driven (stator field) discs are notgive the angle of shift relative to each other. When the appearance braking torque (stator response) shear angle between discthe greater the braking torque, the greater the mi will be. Obviously, with an increase in the braking torque, one disk can rotate relative to the other, which is a violation of the stability of the system under consideration.

The power of the turbine depends on the amount of energy carrier, and in coordinates R,δ is represented by a straight line.

At certain values ​​of the generator EMF E and voltagereceiving systemU the power characteristic has a maximum, which is calculated by the formula

This is sometimes referred to as the “ideal” power limit for the simplest electrical system. The specified value of the turbine power corresponds to two points of intersection of the characteristicsa and b (Fig. 9.2, a), in which the power of the generator and turbine equalinnovate each other.

Consider the mode of operation at the point a. If the power of the generatorpa for some reason will change by the value ΔР, then the angle δ,following a sinusoidal dependence, will change by Δδ. From fig. 9.2,ait follows that at the point a a positive power increment corresponds to a positive angle increment.

When changing the generator power, the equilibrium of the moments turbine and generator is disrupted. With an increase in the power ge the generator on the shaft connecting it with the turbine, there is precise braking torque, since the braking torque of the generator dominates the torque of the turbine. Underthe influence of the braking torque, the generator rotor begins to deceleratechange, which causes the movement of the rotor and the associated eyelidtorus EMF E in the direction of decreasing the angle δ (Fig. 9.2, b). It should be emphasized that the movement of the rotor under the influence of excessive

font-size: 9.0pt; color: black "> Fig. 9.2. K determination of the criterion of statistical stability of the simplest system themes: a - power characteristic; b - deviation of the EMF vector from the statebalance; v - falling out of synchronicity; G - mechanical interpretation

moment is superimposed on its movement in a positive direction with a synchronous speed, which is many times higher soonof this movement. As a result, at the point a recoversthe initial mode of operation and, as follows from the definition of static stability, this mode is stable. Samethe conclusion can also be obtained with a decrease in the power of the generator inpoint a. At the pointb a negative increment in the generator power corresponds to a positive increment in the angle.

With a decrease in the power of the generator, a mustache appears on the shaftgross excess torque that increases the angle d. WITH an increase in the angle, the power of the generator decreases, this increases the accelerating moment, i.e., an avalanche-like process arises, calledout of synchronicity. The process of falling out ofsynchronism and asynchronous mode, in which it ends upgenerator, characterized by the continuous movement of the vectorEMF E relative to voltageU receiving system (fig.9.2, v).

If at the point b excessive braking torque occurs(the generator power will increase), then it will cause movementoperating point of the turbine-generator system to the point a.

Many fundamental issues of electromechanical transients are considered using simple circuits of electrical power systems. These schemes are called power system models, moreover, the word "model" is often omitted, but necessarily implied, since any scheme of a power system is essentially a model of this power system.

The most common one-machine, two-machine and three-machine models of power systems. The simplest of which is single-machine model of the power system, which also has a name car-tire model.

The simplest (single-machine) model of the power system is represented by one remote power plant (equivalent generator) operating through transformer connections and a power line in parallel with the generators of a powerful concentrated power system, so powerful that its receiving buses are designated as infinite power buses (BWM). Distinctive features of the BWM are voltage constant in modulus (U = const) and constant frequency (about 0 = const of this voltage. electrical diagrams are usually not depicted. In equivalent circuits, infinite busbars are used as an element representing a powerful system.

Consider the processes in a single-machine power system (Fig. 1.2, a), in which from a remote unregulated generator G through transformers T | and T 2 and a single-circuit power line L, active power will be transmitted R at current / into the power system C. Power is supplied to the receiving buses of the power system, which are taken as buses of infinite power. Let us determine the basic relationships between the parameters of the mode of a single-machine power system, which are necessary for the analysis of processes.

Let us assume, by way of simplification, that the active resistances and admittances of all elements of the system are equal to zero (r = 0; g = 0; b = 0), and draw up an equivalent circuit. Under these assumptions, the equivalent circuit has the form of a chain of inductive resistances (Fig. 1.2, b), connected between two sources of electromotive forces (EMF). Source E the synchronous EMF of the generator is simulated by the source U- voltage on the BWM.

Rice. 1.2. Single machine power system model

Equivalent inductive reactance NS v equivalent circuit substitution (see Fig. 1.2, c) is defined as the sum of inductive reactances:

Relationship between power R, modules E, U vectors E q, U and the angle 5 between them will be determined using a vector diagram of voltages, emf and currents (Fig. 1.3), acting in an equivalent equivalent circuit.

The diagram shows the active and reactive / p components of the current / and, respectively, shows the longitudinal Ljx and transverse I ^ jx voltage drop components / jx on equivalent resistance NS. EMF E q f and voltage (UV are represented by phase quantities.

It follows from the diagram that the modulus of the transverse component / jx will be determined by the ratio

Multiplying both sides of this equality by 3? / F / x, we obtain where E, U- modules of the corresponding linear quantities.

Rice. 1.3.

power systems

Considering that the three-phase power is defined as P = 3? / F / a, we represent the last equality in the form of the dependence

At E q - const, U= const dependence (1.22) is

sinusoidal function of the active power of the generator from the angle. The graphical representation of this function is called the angular characteristic of the active power of the generator. This name is retained for graphical depictions of dependencies. P (b) and more difficult cases, for example, with changing parameters E (/, U or when the generator is operating as part of a complex power system.

To consider the concept of static stability, a graphical representation of the segment of the function is required R( b) within the positive half-period of the sinusoid (Fig. 1.4).

The angular characteristic is the locus of points corresponding to all possible values ​​of the power transmitted from the generator. In the steady state, only one specific value of power is transmitted from the generator, which corresponds to a specific value of the angle. This power P 0 equal to the power of the turbine R t, as a result of which the turbine, shaft and rotor of the generator maintain a uniform rotary motion.

Rice. 1.4.

Thus, in a steady state, two identical in terms of absolute value, but opposite in the direction of the torque: the accelerating mechanical moment of the turbine and the braking electromagnetic moment of the generator. The analogs of these moments used in the electric power industry are the mechanical power of the turbine R T and electric power of the generator P 0(see fig. 1.4). The deviation of any of these powers (moments) from the steady-state value is reflected in the form of an imbalance of powers (moments) AR = P T - P on the shaft, under the action of which the generator rotor will accelerate or decelerate its rotational motion. Accordingly, the value of the angle 5 will increase or decrease.

As seen in Fig. 1.4, there are two points of intersection (a and B) turbine characteristics P t and angular characteristics R( 5) generator. The question arises about the possibility of stable work at each of these points.

Let us assume that the steady-state mode of the generator is characterized by the point a. With a random increase in the generator power by an amount AR a and a corresponding increase in the angle by the value D8 ((the equality of the moments acting on the shaft will be violated, and the braking electromagnetic moment of the generator will be greater than the accelerating moment of the turbine. Under the influence of the excess braking torque, the rotor movement will begin to slow down, accompanied by a decrease in the angle and the active power of the generator supplied to the network. will continue until ns the equality of the accelerating and decelerating torques is restored, that is, until the system returns to the initial mode, characterized by the point a.

Thus, when working at the point a the power system mode is statically stable, since the system is able to return to its original state under the action of small disturbances.

When working at a point b a slight increase in the angle is accompanied by a decrease in the active power supplied to the network. When you accidentally jump to a point B " the power of the turbine will be greater than the power of the generator by the value AP h. Accordingly, the accelerating mechanical moment of the turbine will be greater than the braking electromagnetic moment of the generator, as a result of which the rotor of the generator will accelerate. This will lead to an increase in angle 8 and, as a consequence, to an increase in the power (moment) imbalance. AR. Further development the process has an avalanche-like character and ends with the remote generator falling out of synchronicity with the generators of the receiving power system.

Thus, the state of the power system corresponding to the point B, is unstable, although at this point, as well as at the point a, there is an equality of the braking and accelerating moments acting on the rotor shaft of the generator.

In practical calculations, criteria (conditions) are widely used, under which the static stability of the power system is preserved. One of these criteria can be easily established with a deeper analysis of stable and unstable regimes. Continuing the reasoning, we note that all points of the angular characteristic located on its ascending branch correspond to stable modes of the power system under consideration. The extreme point is frustrating of the ascending and descending branches of the characteristic and, therefore, is the boundary point. It is generally accepted to refer this point to the region of stable regimes.

At any point in the ascending branch of the angular characteristic, a randomly occurring power imbalance AR and the corresponding increment of angle D5 have the same signs, their ratio is positive and can be considered as a formal sign of stability

When passing to infinitely small increments and taking into account the extreme point of the angular characteristic, where dP / d8 = 0, this feature is written as

and used like a practical criterion for the static stability of a single-machine power system.

Derivative dP / d8 called sync power... It can be calculated using the formula

The power system mode limiting according to the conditions of static stability corresponds to the equality

In this mode, the limiting angle is 5 pr = 90 °, and the limiting, that is, the maximum possible, transmitted power R m defined as

Obviously, under operating conditions, the generator should not be loaded to its maximum capacity. R m, since any slight deviation of the mode parameters can lead to a loss of synchronism and the transition of the generator to asynchronous mode. In case of unforeseen disturbances, a generator load margin is provided, characterized by static stability safety factor

The guidelines for the sustainability of power systems prescribe that in normal modes, a margin corresponding to the factor K st> twenty %. In the most severe modes, in which an increase in power flows along the lines allows to reduce the number of consumers or the loss of water resources, it is allowed to reduce the stability margin to K sg> 8%. In short-term post-emergency modes, a reserve must also be provided To st> 8%. In this case, iodine is understood as short-term nasal-emergency modes lasting up to 40 minutes, during which the dispatcher must restore the normal margin of static stability.

The main task of the electric power industry is uninterrupted, sustainable supply of the consumer electrical energy... It is necessary to determine under what conditions it is possible to ensure the stable operation of generators, what amount of power can be transmitted through the power line, what factors ensure stability, why is the stable, parallel operation of synchronous generators in normal operation disturbed. Let's start considering these issues.

Fig 7. The simplest scheme electrical system

For the presented power transmission scheme in the previous section, an expression was obtained for the electric power as a function of the angle between the vectors of the emf. Eq and voltage of the receiving buses U, which is called the angular characteristic:

For given values ​​of Eq, U, Xd, the generator power is a function of the angle, and this dependence is nonlinear - sinusoidal. For completeness, the power characteristic of the PT turbine is drawn on the same graph, and since it does not depend on the angle, it is represented by a straight line.

Rice. eight.

The balance of powers on the generator shaft, i.e. synchronous operation is provided at Pg = PT, i.e. when the rotating mechanical power(torque) of the turbine and the braking electromagnetic power (torque) of the generator. This statement also follows from differential equation the relative motion of the rotor of a synchronous machine, discussed in the previous lecture

at Pg = PT, = constant. (21)

As can be seen from the graph in Fig. 8, the condition PG = PT is fulfilled at two points 1 and 2, which correspond to angles 1 and 2. It is necessary to determine at which of these points the generator will operate stably.

Suppose that as a result of some action, the angle at point 1 deviated by a small amount. In this case, the electromagnetic power of the generator and the power transmitted through the power line increased by the value P1, while the mechanical power of the turbine did not change due to inertia. The condition for the balance of powers (moments) on the shaft was violated, since (Pg1 + P1)> PT, and the braking torque prevails on the shaft, under the action of which the rotor of the generator is decelerated. As a result, the angle begins to decrease to 0, and the rotor returns to point 1, where the moment equilibrium is ensured. A similar process - a return to point 1 occurs if the angle at this point decreases by.

If the same increase in the angle by an amount occurs at point 2, then the excess torque arising on the shaft will be accelerating, since (Pg2 - P2)

Consequently, from two points 1 and 2, the mode at point 1 is stable, since the rotor returns to the starting point with small deviations. Therefore, a sign of the stability of the synchronous generator is the return to its original mode. It must be remembered that the restoration of the original mode or close to it is the main indicator of the stable operation of the synchronous generator and, accordingly, the electrical system.

As the turbine power increases and, accordingly, the power transmitted through the line according to the graph, the angle value also increases, approaching point 3. This point, on the one hand, shows the maximum active power of the generator that can be transmitted at m = 900:

where Pm = is the maximum power. On the other hand, the point is the boundary point separating the stable and unstable areas of the generator.

It must be remembered that the limits of the angle change:

0900 is the zone of stable operation of the synchronous generator;

-> 900 area of ​​unstable operation of the synchronous generator.

The maximum power Pm = is called the ideal static limit of the transmitted power, corresponding to the constant voltage U, which is not always the case.

In practical calculations, in order to quantify the level of static stability (stability with small deviations), the concept of a safety factor for static stability is introduced, determined by the following ratios:

The Kc value is set within the range of not less than:

20% in normal modes,

8% in post-emergency modes.

It was found that the stable operation of the synchronous generator is ensured if the signs of the increments of the angle and power P = PT ± Pg coincide. Then for deviations you can write:

or, passing to the derivative:, since PT = post.

Thus, static stability will be ensured if the condition

This condition is mathematical criterion static stability of a synchronous machine. The problem and essence of stability under small perturbations are reduced to the adoption of measures under which this condition will be satisfied. They will be discussed further.

It should be noted once again that the possibility of transferring active power through a power transmission line is associated precisely with the presence of a shift angle between the vectors of the emf. Eq and the voltage of the receiving system U, in other words, the shear angle between the voltage vectors at the ends of the transmission. Thus, a change in the intake of an energy carrier (steam or water) into the turbines of a transmission station and their mechanical power is reflected in electric mode transmission by changing the angle, which is a quantity that characterizes both the stability of the transmission and its limiting mode.

Measures to ensure a margin of static stability of the electrical system

In order to avoid violations of the static stability of the electrical system, the following conditions must be met:

The maximum power transmitted through power lines should not exceed the maximum permissible values, which is tantamount to setting the limiting angles of displacement of the rotors of generators;

Stress levels, especially at load nodes, should not fall below the permissible level.

These conditions are ensured both during the operation of the electrical system and in the process of its design with the selection of appropriate equipment, since their parameters must be selected based on these requirements.

The value of the margin of static stability due to the above conditions has a significant practical significance, and its provision and increase depend on many factors.

Let's consider the most important of them.

Let a simple diagram of an electrical system be given

Fig 9 The simplest diagram of an electrical system.

Fig 10. Electrical system equivalent circuit

The power transmitted from the generator is determined by the expression:

In case of neglect of active resistances of elements electrical network(ri = 0) this formula is simplified

It can be seen from the structure of the formula that by acting or changing the values ​​included in Pm, it is possible to increase the maximum characteristic or, which is the same, to increase the maximum transmitted power and thereby increase the margin of static stability, determined by the ratio:

Let's consider them separately and determine the possibilities of changing them. Let's start with inductive reactances.

Resistance. Resistances of transformers and their change are associated with design features apparatus, therefore, during the period of operation, a working transformer in the calculations of static stability is represented by a given resistance determined by the nominal data: power, short-circuit voltages of the steps, etc. The resistances of the power lines included in the formula can change in the event of disconnection of one of the circuits, part and section. Since Xl is included in the denominator of the power expression, respectively, the maximum of the angular characteristic changes: when one of the circuits is disconnected, its value decreases from Pm1 to Pm2, and the angle value corresponding to the normal mode increases from 1 to 2. In order to increase Pm, a new circuit is added.

Fig 11.

It should be noted that increasing the number of parallel circuits of the power transmission line in order to increase the maximum transmitted power and the margin of static stability is an expensive measure. Therefore, in long lines, they use (in addition to the transition to a higher voltage class) the splitting of the phase wires of the power transmission line. As you know, the specific inductive resistance of the line, referred to 1 km, is determined by:

where Dav is the geometric mean distance between the phase wires, re is the equivalent radius.

The decrease in the inductive resistance of the line when splitting the phase wires is explained by the redistribution of the magnetic fields of the wires: the fields between the split wires are weakened and forced outward, as if increasing the cross-section of the wire at the same metal consumption. It should be noted that each additional wire, as it splits, gives less and less additional effect... For example, with two wires in a phase, the inductive resistance decreases by 19%, with three wires by 28%, with four wires by 32%, etc.

The values ​​of the specific inductive resistances during splitting vary from 0.410.42 ohm / km to 0.26 0.29 ohm / km. The phase wire is split into two, three, four and more wires connected in parallel. For example, with a line voltage of 330 kV - 2 wires per phase, 500 kV - 3 wires, 750 kV - 5 wires and 1150 kV - 8 wires per phase. Therefore, such a measure leads to an increase in the maximum transmitted power without increasing the consumption of the wire material, since its total cross section does not increase.

Taking into account the load with a constant resistance increases the total resistance and therefore reduces the maximum characteristic.

The synchronous generator has the highest inductive resistance.

There is a certain relationship between the values ​​of the parameters of machines and their cost, since inductive resistances are determined by the values ​​of electromagnetic loads. Reducing the inductive reactances of a synchronous generator, especially Xd, is an extremely difficult and expensive path associated with an increase in the size of the machine and a decrease in the coefficient useful action... Let's consider this issue in more detail.

As you know, the values ​​of synchronous inductive reactances are inversely proportional to the size of the air gap of the machine.

where is the air gap.

At the same time, Xd is also inversely proportional to the excitation current

From these relations it can be seen that in order to reduce the synchronous inductive resistance, it is necessary to increase the air gap and the excitation current, which is necessary to create an additional magnetic flux providing increased energy processes. Consequently, in this case, it becomes necessary to increase the excitation power, to strengthen the excitation winding and other windings, which is associated with an increase in the consumption of material. Due to the difficulty in placing the excitation winding, this will lead to an increase in the size of the generator. Therefore, in general, a decrease in Xd and Xq will lead to an increase in the cost of the car.

A decrease in the transient inductances Xd ", Xq" of a synchronous generator is possible due to an increase in the current density in the winding, which leads to an increase in losses, a decrease in efficiency, an increase in the weight of the generator and, accordingly, in the cost of the generator.

These problems are of particular importance in the creation of modern, highly used synchronous generators with a capacity of 200-1200 MW.

The use of ARVs is more effective different types, with the help of which, in essence, the synchronous and transient inductances of the generators are compensated.

Change in emf generator (in in this case Eq) leads to a change in two important parameters: its power factor and the voltage on the tires of the machine. Modern highly used synchronous generators are manufactured with high values ​​of the rated power factor cos = 0.9-1. An increase in the rated power factor, at a given active power, leads to a decrease in the rated reactive power, dimensions and cost of the generator, since this reduces the total power of the machine () and, consequently, the consumption of active and structural material will be less. On the other hand, an increase in cоs leads to a decrease in the emf. Eq, which reduces the static stability margin. In addition, the economically optimal transmission length of the reactive power generated by the generator is limited by the distance (25-70) km. The reactive power required for the load must be generated at the point of consumption.

The change in the voltage of the generator depends on its load and to maintain it at the required level, for example, nominal, in a wide range of load changes, it is necessary to change the emf. generator by changing its excitation current. This problem is successfully solved by various types of ARV, which essentially compensate for the internal resistance of the generator.

For example, in the presence of ARV-s, the internal resistance of the synchronous generator to the buses of the starting end, including the resistance of the XT1 transformer, can be compensated by appropriate regulation of the generator excitation, which ensures the constant voltage UГ = const. The maximum of the angular characteristic in this case can be determined from the relation

For comparison, the angular characteristics for various types of ARV are shown (Fig. 12)

Fig 12

As can be seen from the active power formula (28), its value is determined by the product of the emf generator and system voltage, or more general view depends on the square of the voltage. Therefore, in a first approximation, we can assume that a doubling of the line voltage is equivalent to an increase in the number of transmission circuits by a factor of four. It follows that increasing the transmission voltage to increase the maximum transmitted power is more economical than increasing the number of transmission circuits.

Longitudinal and lateral compensation of the parameters of the power transmission line are also measures to increase the maximum transmitted power and increase the static stability margin.

Longitudinal compensation means a series connection of capacitors in the line, in which the resistance value decreases from Chl to (Chl-Xc) where Xc is the capacitive resistance of the capacitor. This measure is especially effective with long power lines.

Lateral compensation is a synchronous compensator connected to the transmission line through a transformer. By maintaining the voltage at the point of connection, the SC essentially has the effect of reducing the line length and, accordingly, its resistance. Currently, very effective, high-speed static sources of reactive power (SIRM) with a response time of (0.02h0.06) sec are used.

These devices have a regulated reactor and an unregulated capacitor, as well as a control system. They, in addition to increasing power, perform a wide range of tasks, carry out phase-by-phase regulation of mode parameters, suppress overvoltage, regulate voltages in a wide range, and increase the margin of static and dynamic stability.

The family of compensators also includes adjustable and unregulated reactors that compensate for the capacitance of power lines and maintain the voltage at the connection point due to the non-linear characteristic of the core saturation.

It should be recalled once again that the criterion for the static stability of the synchronous generator is the condition and at the maximum transmitted power Pm the synchronizing power becomes equal to zero.

Therefore, in practical conditions it is impossible to transmit this power, since the slightest shock of the load in the EES causes the generator to fall out of synchronism, therefore, the normal transmitted power P0 must be less than Pmax. And its value will be determined based on the safety factor of the static stability of the system.

From the above, we can conclude the following:

The ideal transmit power limit is the maximum power delivered to the system assuming a constant voltage on the receiving end buses.

The criterion for the static stability of the simplest system is the positiveness of the derivative of the transmitted power with respect to the angle between the emf of the generators and the voltage of the receiving end of the transmission.

The static stability safety factor shows by what amount the transmitted power from the station to the network can be increased in order to prevent a violation of the stability of the electrical system.

4. Modern automatic excitation regulators (ARV-s, ARV-p) can compensate for the inductive resistances of elements, including the inductive resistances of a synchronous generator, due to effective regulation of the excitation system depending on the parameters of the electrical system mode.

Evaluating all of the above measures to increase the static power limit, it can be concluded that the most economical measures are those aimed at maintaining a constant voltage at the terminals of generators and on the load buses. The use of various types of ARVs on generators and modern high-speed static sources of reactive power is practically the most rational and economic measure to increase the limits of the transmitted power and the static stability margin, both for an individual transmission and for the electrical system as a whole.

abstract

Explanatory note contains 21 pages, 6 tables, 14 figures, 3 sources of literature, in which the calculation methodology that was used in this work is described in detail.

Research object: power transmission system.

Purpose of the work: to get the skills to calculate electromechanical transients in the power transmission system, to calculate the maximum voltage drop on the buses of an induction motor, to evaluate the static and dynamic stability of the system.

Introduction

Initial data

Conclusion

Introduction

Power system stability- this is its ability to return to its original state with small or significant disturbances. By analogy with mechanical system the steady state of the power system can be interpreted as its equilibrium position.

Parallel operation of generators of power plants included in the power system differs from the operation of generators at one station by the presence of power lines connecting these stations. The resistances of the power lines reduce the synchronizing power of the generators and make it difficult for them to work in parallel. In addition, deviations from the normal operation of the system, which occur during outages, short circuits, sudden load shedding or surges, can also lead to a breakdown in stability, which is one of the most severe: accidents leading to an interruption in the power supply of consumers. Therefore, the study of the problem of stability is very important, especially when applied to AC power lines. There are two types of stability: static and dynamic.

Static stability is the ability of a system to independently restore the original mode under small and slowly occurring disturbances, for example, with a gradual insignificant increase or decrease in the load.

Dynamic power system stabilitycharacterizes the system's ability to maintain synchronism after sudden and drastic changes parameters of the mode or in case of accidents in the system (short circuits, disconnections of the frequency of generators, lines or transformers). After such sudden disruptions in normal operation, a transient process occurs in the system, after which the established post-emergency mode of operation should again occur.

It is these sudden disruptions in the operation of the SES that lead to severe economic consequences for the population and industrial facilities.

Modern energy sector pays very much great attention the fight against accidents on lines, short circuits, makes a great contribution even at the design stage of the SES of cities and enterprises.

Initial data

The diagram for the calculation is shown in Figure 1.

Figure 1 - Diagram of the power transmission system

The initial data for calculating the first and second tasks are taken from the table in accordance with the option number.

Technical data of transformers:

Transformer type,

MVA Regulatory limits

vaniya,%, kV

windings,%

% VNTDC-250000 / 110250-11013.8; 15.75; 1810.56402000.5TDC-630000 / 110630-1102010.59003200.45

Parameters of double-circuit overhead power line

Wire brand,

Ohm / km Length

l, kmU, kV AS-3300.1070.3670.3820.3301.3890.931300110

Figure 2 - Scheme of the system for calculating the limiting voltage drop on the buses of an induction motor

The initial data for calculating the third problem are taken below in the table in accordance with the variant number.

Technical details asynchronous motor

Type Rated data Starting characteristics P, kW I, AN, rpm , %, kg * m 2U, kVn 0, rpm DAZO 17-39-8 / 1050061.574191.00.855.20.652.12886741

CL parameters:

Wire type Length l, kmx 0, Ohm / km APvV 1 * 3000,0350,099

We draw up the equivalent circuit of the system, which is shown in Fig. 1 and calculate the inductive resistance of all elements:

Figure 3 - Equivalent circuit of the system

inductive reactance given,

inductive reactance of transformers:

inductive resistance of power lines:

All resistances of the equivalent circuit are reduced to the rated voltage of the generator. Resistance of transformers:

power line resistance:

Determine the total resistance of the system:

We calculate the rated reactive power of the generator:

We determine the approximate value of the synchronous EMF of the generator:

Determine the value of the static stability safety factor:

Based on the calculation data, we build a vector diagram.

Figure 4 - Vector diagram

The calculation results are entered in table 3.

Table 3

MW0162312.5442541603.7625603.7541442312.51620

Figure 5 - Angular power characteristic

The system is statically stable since the safety factor is greater than 20%. And the limit of the transmitted power of the generator to the system is reached at coal? = 900.

We calculate the modes in turn.

2.1 Calculation of emergency and post-emergency operation for single-phase short circuit at point K-1

1.1 Normal mode

1.2 Emergency mode

We draw up the equivalent circuit of the system with a single-phase short circuit

Figure 6 - Equivalent circuit for emergency mode with single-phase short circuit

Total short-circuit resistance X ?with a single-phase short circuit is equal to the sum of the negative sequence resistance and zero sequence resistance.

We transform the equivalent circuit of the system with a single-phase short circuit from a "star" connection to a "delta" connection with sides X 1, X 2, NS 3.

Resistance X 2 their 3 can be discarded because the power flow delivered by the generator to the network does not pass through these resistances.

Figure 7 - Converted equivalent circuit

Let's determine the total resistance of the system:

Where X? = X2? + X0? - an asymmetrical short-circuit shunt, which is connected between the beginning and the end of the positive and negative sequence.

Determine the inductive resistance of the zero sequence X0 ?:

Determine the inductive reactance of the negative sequence X2?

Determine the resistance of the shunt short circuit X ?:

X2? + X0? = 3 +0.097 = 3.097 Ohm

Xd? II = 20.2 + 0.1 + 3.5 +0.04 + = 47Ω.

We determine the limit of the transmitted power of the generator to the system:

By changing the values ​​of the angle from 0 to 180 degrees, we calculate the corresponding values ​​of the power supplied by the generator to the system according to the formula:

The calculation results are entered in table 4.

Table 4

Gra, MW 081.3157222.3271.9303.3314303.3271.9222.315781.30

1.3 Post-emergency mode

We draw up the equivalent circuit of the system for the post-emergency mode.

Figure 8 - Equivalent circuit for the post-emergency mode with a single-phase short circuit

The post-emergency mode is determined by disconnecting one power line circuit, after which the resistance changes:

Determine the total resistance of the system:

We determine the limit of the transmitted power of the generator to the system:

We calculate the value of the angles:

Totkl = +

Since the line is protected, after a while it will be disconnected by switches. Therefore, we choose a VGBE-35-110 series SF6 circuit breaker with a trip time = 0.07 s. Short circuit relay protection devices should also be provided. We select the current relay RT-40 with setting time = 0.08 s.

0.07 + 0.08 = 0.15 s,

We find the time for switching off the short circuit:

Totkl = 0.07 + 0.15 = 0.22 s.

29? 0.22, which satisfies the condition? Totkl

By changing the values ​​of the angle from 0 to 180 degrees, we calculate the corresponding values ​​of the power supplied by the generator to the system according to the formula:

Table 5

The calculation results are entered in table 5.

hail 0153045607590105120135150165180,

MW0 140 270.5382.5468.5522.6541522.6468.5382.5270.51400

We build in one coordinate plane angular characteristics of power in normal, emergency and post-emergency modes, on the graph we indicate the value of the turbine power P 0... Taking into account the calculated value of the limiting angle of short circuit breaking ?off plot the areas of acceleration and deceleration on the graph.

Figure 9 - Graph of angular characteristics of powers and areas of acceleration and deceleration at a single-phase short circuit

2.2 Calculation of emergency and post-emergency mode with a three-phase short circuit at point K-2

2.2.1 Normal Mode

The calculation of the normal mode was carried out in problem 1.

2.2 Emergency mode

We draw up the equivalent circuit of the system with a three-phase short circuit

Figure 10 - Equivalent circuit of the system with a three-phase short circuit

With a three-phase short circuit at point K-2, the mutual resistance of the circuit becomes infinitely large, because shunt resistance short-circuit X ? (3) = 0. In this case, the power characteristic of the emergency mode coincides with the abscissa axis.

2.3 Post-emergency mode

The equivalent circuit for a three-phase short circuit and the calculation of the post-emergency mode is similar to the post-emergency mode given in clause 2.1.3

We calculate the value of the angles:

We find the limiting angle of switching off the short circuit? Off:

We calculate the limiting time for switching off the short circuit:

We select the appropriate settings for the operation of relay protection devices:

Totkl = +

Since the line is protected, after a while it will be disconnected by switches. Therefore, we select the series SF6 circuit breaker

VGT - 110 with shutdown time = 0.055 s. Short circuit relay protection devices should also be provided. We select the current relay RT-40 with setting time = 0.05 s.

The duration of the relay protection is determined by:

0.005 + 0.05 = 0.055 s,

We find the time for switching off the short circuit:

Totkl = 0.055 + 0.055 = 0.11 s.

17? 0.11, what satisfies the condition? Totkl

We plot the angular characteristics of power in one coordinate plane in normal, emergency and post-emergency modes, on the graph we indicate the value of the turbine power P0. Taking into account the calculated value of the limiting angle of switching off the short circuit? Off, we plot the areas of acceleration and deceleration on the graph.

Figure 11 - Graph of angular characteristics of powers and areas of acceleration and deceleration at a three-phase short circuit

To determine the dynamic stability of the system with a single-phase short circuit, it is necessary to consider the areas of acceleration Fstart and braking Fbrake. The condition for the dynamic stability of the system is the inequality: Fusk? Ftorm. It can be seen with the naked eye from the graph of the angular characteristic that the acceleration area is an order of magnitude larger than the braking area, which means that the system is not dynamically stable. Consequently, the accumulated kinetic energy does not have time to turn into potential energy, as a result, the rotor speed and angle? will grow and the generator will fall out of synchronicity. To determine the static stability of the system, it is necessary to find the safety factor. Having calculated the safety factor, we can conclude that the system is statically stable, since.

We calculate the parameters of the transmission elements and the load parameters reduced to the base voltage U b = 6 kV and base power:

Sb = SAD nom =,

Line resistance:

Leakage inductive reactance of the magnetic circuit of the motor:

Determine the active power consumed in the initial mode of the engine:

We find the active resistance of the motor rotor in the initial mode (simplified equivalent circuit of an induction motor):

0392 +0,05? = ,

replace with x and get:

05x2 - x + 0.0392 = 0;

D= B2 - 4ac = 12 - 4? 0.05? 0.0392 = 0.99216;

We choose the largest of the roots of the equation and get:

Determine the reactive power consumed in the initial mode by the engine:

Determine the voltage on the system buses in the initial mode:

Determine the voltage on the tires of the system at which the engine is braked:

Determine the margin of static voltage stability of the motor:

To construct a mechanical characteristic M = f (S) according to the equation

M =, you need to make the following calculation:

Determine the nominal rotor speed:

nom = n0? (1 - Snom) = 741? (1-0.01) = 734 rpm.

Find the critical slip:

cr = Snom? (?? +) = 0.01? (2.1 +) = 0.039.

Determine the rated and maximum (critical) moments of the engine:

Mnom = = N? M,

Мmax = ?? ? Mnom = 2.1? 6505.3 = 13661.4 N? M.

To construct a mechanical characteristic, we use the Kloss formula:

Having asked different meanings slip S, we find the corresponding values ​​of the moment M. The calculation results are entered in table 6.

Table 6

SM, N? M000.0166480.039136610.06124190.08105890.192620.251260.335020.426420.521180.617630.715180.813320.9115011064

According to table 6, we build a graph M = f (S):

Figure 12 - Graph of the mechanical characteristics of an induction motor

The system is statically stable since the voltage safety factor of the motor is greater than 20%

Conclusion

After completing this term paper theoretical knowledge acquired during the semester by calculation was worked out and consolidated different types KZ; checking the system for static and dynamic stability; construction of angular characteristics of power and mechanical characteristics of asynchronous.

I learned how to analyze the system for stability, calculate the modes of operation of the system before, after, and during various types of short circuit.

It can be concluded that the calculation of electromechanical transients occupies one of the significant positions in the calculation and design of various simple and complex power supply systems.

Bibliography

1. Kulikov Yu.A. Transient Processes in Electrical Systems: Textbook. allowance. - Novosibirsk: NSTU, M .: Mir: OOO "AST Publishing House", 2008. -

Borovikov V.N. and others. Electric power systems and networks - Moscow: Metroizdat., 2010. - 356 p.

Apollonov A.A. Calculation and design of relay protection and automation - St. Petersburg, 2009. - 159 p.

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