Power system (power system). Electric power (electrical) system

Within the limits of electrostatics, it is impossible to give an answer to the question of where the energy of a capacitor is concentrated. The fields and charges that formed them cannot exist in isolation. They cannot be separated. However, alternating fields can exist independently of the charges that excited them (radiation from the sun, radio waves, ...), and they transfer energy. These facts force us to admit that the carrier of energy is an electrostatic field .

When electric charges move, the forces of the Coulomb interaction perform a certain work d A... The work done by the system is determined by the decrease in the interaction energy -d W charges

Energy of interaction of two point charges q 1 and q 2 at a distance r 12, is numerically equal to the work of moving the charge q 1 in the field of a stationary charge q 2 from a point with a potential to a point with a potential:

It is convenient to write down the interaction energy of two charges in a symmetric form

.

(5.5.3)

For a system from n point charges (Fig.5.14) due to the principle of superposition for the potential, at the point of location k th charge, you can write:

Here φ k , i- potential i th charge at the point of location k th charge. The sum excludes the potential φ k , k, i.e. the effect of the charge on itself, which is equal to infinity for a point charge, is not taken into account.

Then the mutual energy of the system n charges is equal to:

(5.5.4)

This formula is valid only if the distance between the charges significantly exceeds the dimensions of the charges themselves.

Let's calculate the energy of a charged capacitor. The capacitor consists of two, initially uncharged, plates. We will gradually subtract from the lower plate the charge d q and transfer it to the top plate (fig. 5.15).

As a result, a potential difference arises between the plates.Each portion of the charge transfers an elementary work

Using the definition of capacity, we obtain

The total work spent on increasing the charge of the capacitor plates from 0 to q, is equal to:

This energy can also be written as

1. First, consider a system consisting of two point charges 1 and 2. Let us find the algebraic sum of the elementary workings of the forces f 1 and F 2 with which these charges interact. Let in some K-frame of reference in time dt charges have made displacements dl 1 and dl 2. Then the work of these forces is δА 1,2 = F 1 dl 1 + F 2 dl 2. Considering that F 2 = -F l(according to Newton's third law): δА 1,2 = F 1 (dl 1 - dl 2). The value in brackets is the movement of charge 1 with respect to charge 2. More precisely, this is the movement of charge 1 in the K "-system of reference rigidly connected with the charge 2 and moving along with it translationally with respect to the original K-system. Indeed, the displacement dl 1 of charge 1 in the K-system can be represented as displacement dl 2 K "-system plus displacement dl 1 charge 1 relative to this K "-system: dl 1 = dl 2 + dl 1. Hence dl 1 -dl 2 = dl` 1 and δА 1,2 = F 1 dl` 1. Work δA1,2 does not depend on the choice of the original K-system The force F 1 acting on charge 1 from the side of charge 2 is conservative (as a central force). Therefore, the work of this force on displacement dl` 1 can be represented as a decrease in the potential energy of charge 1 in the field of charge 2 or as a decrease in the potential energy of interaction of this pairs of charges: δА 1,2 = -dW 1,2, where W12 is a value that depends only on the distance between these charges.

2. Let us turn to a system of three point charges (the result obtained for this case can be easily generalized to a system of an arbitrary number of charges). The work done by all the forces of interaction during elementary displacements of all charges can be represented as the sum of the work of all three pairs of interactions, i.e. δА = δA 1,2 + δA 1,3 + δА 2,3. But for each pair of interactions δA i, k = -dW ik, therefore δА = -d (W 12 + W 13 + W 23) = - dW, where W is the interaction energy of the given system of charges, W = W 12 + W 13 + W 23. Each term of this sum depends on the distance between the corresponding charges, therefore the energy W of a given system of charges is a function of its configuration. Similar reasoning is valid for a system of any number of charges. Hence, it can be argued that each configuration of an arbitrary system of charges has its own energy value W, and δА = -dW.

Energy of interaction... Consider a system of three point charges, for which it is shown that W = W 12 + W 13 + W 23. We represent each term W ik in symmetric form: W ik = (W ik + W ki) / 2, since W ik = W ki. Then W = (W 12 + W 21 + W 13 + W 3l + W 23 + W 32) / 2. Let's group the terms: W = [(W 12 + W 13) + (W 21 + W 23) + (W 3l + W 32)] / 2. Each sum in parentheses is the energy Wi of interaction of the i-th charge with the rest of the charges. That's why:

Bearing in mind that W i = q i φ i, where q i is the i-th charge of the system; φ i is the potential created at the location of the i-ro charge by all other charges of the system, we obtain the final expression for the interaction energy of the system of point charges:

Total interaction energy... If the charges are distributed continuously, then, expanding the system of charges into a set of elementary charges dq = ρdV and passing from summation in (4.3) to integration, we obtain

(4.4), where φ is the potential created by all charges of the system in an element of volume dV. A similar expression can be written for the distribution of charges over the surface, replacing ρ with σ and dV with dS. Let the system consist of two balls having charges q 1 and q 2. The distance between the balls is much larger than their sizes, so the charges q l and q 2 can be considered pointlike. Find the energy W of a given system using both formulas. According to formula (4.3), where φ 1 is the potential created by the charge q 2 at the location of the charge q 1, the potential φ 2 has a similar meaning. According to formula (4.4), it is necessary to split the charge of each ball into infinitesimal elements ρdV and multiply each of them by the potential φ, created not only by the charges of another ball, but also by the elements of the charge of this ball. Then: W = W 1 + W 2 + W 12 (4.5), where W 1 - the energy of interaction with each other of the elements of the charge of the first ball; W 2 - the same, but for the second ball; W 12- the energy of interaction of the elements of the charge of the first ball with the elements of the charge of the second ball. Energy W 1 and W 2 is called the intrinsic energies of the charges q 1 and q 2, and W 12 is the energy of interaction of the charge q 1 with the charge q 2.

The energy of a solitary conductor... Let the conductor have a charge q and potential φ. Since the value of φ at all points where there is a charge is the same, φ can be taken out from under the integral sign in formula (4.4). Then the remaining integral is nothing but the charge q on the conductor, and W = qφ / 2 = Cφ 2/2 = q 2 / 2C (4.6). (Taking into account that C = q / φ).

Capacitor energy... Let be q and φ are the charge and potential of the positively charged capacitor plate. According to formula (4.4), the integral can be divided into two parts - for one and the other plates. Then

W = (q + φ + –q _ φ _) / 2. Since q_ = –q + , then W = q + (φ + –φ _) / 2 = qU / 2, where q = q + - capacitor charge, U- potential difference across the plates. С = q / U => W = qU / 2 = CU 2/2 = q 2 /2C(4.7). Let us consider the process of charging a capacitor as a transfer of charge in small portions dq "from one plate to another. The elementary work performed by us in this case against the field forces is written as d A = U'dq ’= (q’ / C) dq ’, where U’ is the potential difference between the plates at the moment when the next portion of the charge dq is transferred. ”By integrating this expression over q " from 0 to q, we get A = q 2 / 2C, which coincides with the expression for the total energy of the capacitor. In addition, the obtained expression for work A is also valid in the case when there is an arbitrary dielectric between the capacitor plates. This also applies to formulas (4.6).

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Electric energy of the system of charges

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An energetic approach to interaction. The energy approach to the interaction of electric charges is, as we will see, very fruitful in its practical applications, and in addition, it opens up the possibility of looking differently at the electric field itself as a physical reality.

First of all, we will find out how you can come to the concept of the interaction energy of a system of charges.

1. First, consider a system of two point charges 1 and 2. Let us find the algebraic sum of the elementary work of the forces F, and F2, with which these charges interact. Let in some K-frame of reference in the time cU the charges have made displacements dl, and dl 2. Then the corresponding work of these forces

6Л, 2 = F, dl, + F2 dl2.

Considering that F2 = - F, (according to Newton's third law), we rewrite the previous expression: Mlj, = F, (dl1-dy.

The value in brackets is the movement of charge 1 relative to charge 2. More precisely, it is the movement of charge / in / ("- a frame of reference rigidly connected with charge 2 and moving along with it translationally relative to the original / (- system. Indeed, the movement dl, charge 1 in / (- the system can be represented as displacement dl2 / ("- system plus displacement dl, charge / relative to this / (" - system: dl, = dl2 + dl,. Hence dl, - dl2 = dl " , and

So, it turns out that the sum of elementary work in an arbitrary / (- frame of reference is always equal to the elementary work performed by a force acting on one charge, in a frame of reference where the other charge is at rest. - reference systems.

Force F „acting on the charge / from the side of charge 2, conservative (as a central force). Therefore, the work of this force on displacement dl can be represented as a decrease in the potential energy of charge 1 in the field of charge 2 or as a decrease in the potential energy of interaction of the considered pair of charges:

where 2 is a value that depends only on the distance between these charges.

2. Now we turn to a system of three point charges (the result obtained for this case can be easily generalized to a system of an arbitrary number of charges). The work done by all the forces of interaction during elementary displacements of all charges can be represented as the sum of the work of all three pairs of interactions, ie 6L = 6L (2 + 6L, 3 + 6L 2 3. But for each pair of interactions, as soon which was shown, ik = - d Wik, therefore

where W is the interaction energy of a given system of charges,

W «= wa + Wtz + w23.

Each term of this sum depends on the distance between the corresponding charges, therefore the energy W

a given system of charges is a function of its configuration.

Similar reasoning is obviously valid for a system of any number of charges. Hence, it can be argued that each configuration of an arbitrary system of charges has its own energy value W and the work of all interaction forces when this configuration changes is equal to the loss of energy W:

bl = -ag. (4.1)

Energy of interaction. Let us find an expression for the energy W. First, consider again a system of three point charges, for which we have shown that W = - W12 + ^ 13 + ^ 23- We transform this sum as follows. We represent each term Wik in symmetric form: Wik =] / 2 (Wlk + Wk), since Wik = Wk, Then

Let's group members with the same first indices:

Each sum in parentheses is the energy Wt of interaction of the i-th charge with the rest of the charges. Therefore, the last expression can be rewritten as follows:

Generalization of an arbitrary

the obtained expression for a system of the number of charges is obvious, because it is clear that the above reasoning is completely independent of the number of charges that make up the system. So, the interaction energy of the system of point charges

Bearing in mind that Wt =<7,9, где qt - i-й заряд системы; ф,- потен­циал, создаваемый в месте нахождения г-го заряда всеми остальными зарядами системы, получим окончательное выражение для энергии взаимодействия системы точечных зарядов:

Example. Four identical point charges q are located at the vertices of a tetrahedron with an edge a (Fig. 4.1). Find the interaction energy of the charges of this system.

The interaction energy of each pair of charges is here the same and equal to = q2 / Ale0a. There are six such interacting pairs in total, as can be seen from the figure, therefore the interaction energy of all point charges of a given system

W = 6 #, = 6<72/4яе0а.

Another approach to solving this problem is based on the use of formula (4.3). The potential φ at the location of one of the charges, due to the field of all other charges, is equal to φ = 3<7/4яе0а. Поэтому

Total energy of interaction. If the charges are distributed continuously, then, expanding the system of charges into a set of elementary charges dq = р dV and passing from summation in (4.3) to integration, we obtain

where f is the potential created by all charges of the system in an element of volume dV. A similar expression can be written for the distribution of charges, for example, over a surface; for this it is sufficient to replace p by o and dV by dS in formula (4.4).

One may mistakenly think (and this often leads to misunderstandings) that expression (4.4) is only a modified expression (4.3), corresponding to the replacement of the concept of point charges by the concept of a continuously distributed charge. In reality, this is not the case - both expressions differ in their content. The origin of this difference is in a different sense of the potential φ included in both expressions, which is best explained by the following example.

Let the system consist of two balls having charges q and q2 "The distance between the balls is much larger than their sizes, so the charges ql and q2 can be considered pointwise. Let us find the energy W of this system using both formulas.

According to formula (4.3)

W = "AUitPi +2> where, φ [is the potential created by the charge q2 at the place

finding a charge has a similar meaning

and potential f2.

According to formula (4.4), we must break the charge of each ball into infinitesimal elements p AV and multiply each of them by the potential φ, created not only by the charges of another ball, but also by the elements of the charge of this ball. It is clear that the result will be completely different, namely:

W = Wt + W2 + Wt2, (4.5)

where Wt is the energy of interaction with each other of the charge elements of the first ball; W2 - the same, but for the second ball; Wi2 is the interaction energy of the charge elements of the first ball with the charge elements of the second ball. The energies W, and W2 are called the intrinsic energies of the charges qx and q2, and W12 is the energy of interaction of the charge with the charge q2.

Thus, we see that the calculation of the energy W according to the formula (4.3) gives only Wl2, and the calculation according to the formula (4.4) gives the total interaction energy: in addition to W (2, also the own energies IF, and W2. Ignoring this circumstance is often a source gross mistakes.

We will return to this question in § 4.4, and now we will obtain several important results using formula (4.4).

Electrical energy of the system of charges.

Work of the field with polarization of the dielectric.

Electric field energy.

Like all matter, an electric field has energy. Energy is a function of the state, and the state of the field is given by the intensity. Whence it follows that the energy of the electric field is a single-valued function of the strength. Since, it is extremely important to introduce the concept of the concentration of energy in the field. A measure of the concentration of the field energy is its density:

Let's find an expression for. Let us consider for this the field of a flat capacitor, assuming it to be uniform everywhere. An electric field in any capacitor arises during its charging, which can be represented as a transfer of charges from one plate to another (see figure). Elementary work͵ spent on charge transfer is equal to:

where, and the complete work:

which goes to increase the field energy:

Considering that (there was no electric field), for the energy of the electric field of the capacitor we obtain:

In the case of a flat capacitor:

since, is the volume of the capacitor equal to the volume of the field. Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, the energy density of the electric field is equal to:

This formula is valid only in the case of an isotropic dielectric.

The energy density of an electric field is proportional to the square of the strength. This formula, although obtained for a uniform field, is valid for any electric field. In the general case, the field energy can be calculated by the formula:

The expression includes the dielectric constant. This means that the energy density in a dielectric is greater than in a vacuum. This is due to the fact that when creating a field in the dielectric, additional work is performed, associated with the polarization of the dielectric. Let us substitute the value of the electric induction vector into the expression for the energy density:

The first term is associated with the field energy in vacuum, the second - with the work expended on the polarization of a unit volume of the dielectric.

Elementary work͵ spent by the field on the increment of the polarization vector is equal to.

The work on the polarization of a unit volume of a dielectric is equal to:

since, as required.

Consider a system of two point charges (see figure) according to the principle of superposition at any point in space:

Electric field energy density

The first and third terms are associated with the electric fields of charges and, respectively, and the second term reflects the electrical energy associated with the interaction of charges:

The intrinsic energy of charges is positive, and the interaction energy can be both positive and negative.

Unlike a vector, the energy of an electric field is not an additive quantity. The energy of interaction can be represented by a simpler relationship. For two point charges, the interaction energy is:

which can be represented as the sum:

where is the potential of the charge field at the location of the charge, and is the potential of the charge field at the location of the charge.

Generalizing the result obtained for a system of an arbitrary number of charges, we get:

where is the system charge, is the potential created at the location of the charge, all the rest system charges.

If the charges are distributed continuously with the bulk density, the sum should be replaced by the volume integral:

where is the potential created by all charges of the system in an element of volume. The resulting expression matches total electrical energy systems.

Natural natural sources from which energy is drawn to prepare it in the required forms for various technological processes are called energy resources. There are the following types of main energy resources: a chemical energy of the fuel; b atomic energy; into water energy, that is, hydraulic; d solar radiation energy; q wind energy. e energy of ebb and flow; Well geothermal energy. Primary energy source or energy resource coal gas oil uranium concentrate hydropower solar ...

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Lecture number 1.

Basic definitions

Power system (power system)consists of power plants, power grids and electricity consumers, interconnected and connected by a common mode and general management of this mode.

Electric power (electrical) systemIs a set of electrical parts of a power plant, electrical networks and electricity consumers, i.e. it is part of the energy system, with the exception of heating networks and heat consumers.

Electrical networkIs a set of electrical installations for the distribution of electrical energy, consisting of substations, switchgears, overhead and cable power lines.

Electrical substationsIs an electrical installation designed to convert electricity of one voltage or frequency to another voltage or frequency.

Power system characteristics

The frequency at all points of electrically connected networks is the same

Equality of consumed and generated capacities

The voltage in different network nodes is not the same

Benefits of power interconnection

Improving the reliability of power supply

Improving the stability of power systems

Improvement of technical and economic indicators of power systems

Stable power quality

Reducing the required power reserve

The loading conditions of the units are improved due to the equalization of the load curve and the reduction of the maximum load of the power system.

The possibility of a more complete use of the generating capacities of electric power plants appears, due to the difference in their geographical position in latitude and longitude.

Operational management of power systems is carried out by their dispatching services, which, on the basis of appropriate calculations, establish the optimal operating mode for power plants and networks of various voltages.

Energy sources

There are renewable and non-renewable energy sources.

Natural (natural) sources from which energy is drawn to prepare it in the required forms for various technological processes are called energy resources.

There are the following types of main energy resources:

a) chemical energy of the fuel;

b) atomic energy;

c) water energy (i.e. hydraulic);

d) the energy of the sun's radiation;

e) wind energy.

f) the energy of the ebb and flow;

g) geothermal energy.

The primary energy source or energy resource (coal, gas, oil, uranium concentrate, hydropower, solar energy, etc.) enters one or another energy converter, the output of which is either electrical energy, or electrical and thermal energy. If heat energy is not generated, then it is necessary to use an additional energy converter from electrical to heat (dotted lines in Fig. 1.1).

The largest part of the electrical energy consumed in our country is obtained by burning fuels extracted from the bowels of the earth - coal, gas, fuel oil (oil refined product). When they are burned, the chemical energy of the fuels is converted into heat.

Power plants that convert thermal energy resulting from fuel combustion into mechanical energy, and this latter into electrical energy, are called thermal power plants (TPP).

Power plants that operate at the highest possible load for a significant part of the year are called base power plants, power plants that are used only during part of the year to cover the “peak” load are called peak power plants.

ES classification:

1. TPP (IES, TPP, GTS, PGPP)
2. NPP (1-circuit, 2-circuit, 3-circuit)
3. HPPs (dam, diversion)

Electrical part of the ES

Power plants (ES) are complex technological complexes with a total number of main and auxiliary equipment. The main equipment is used for the production, transformation, transmission and distribution of electricity, the auxiliary equipment is used to perform auxiliary functions (measurement, signaling, control, protection and automation, etc.). We will show the interconnection of various equipment on a simplified schematic diagram of an ES with busbars of generator voltage (see Fig. 1).

Rice. 1

The electricity generated by the generator is fed to the busbars of the SS and then distributed between the auxiliary needs of the MV, the generator voltage load of the NG and the power system. Individual elements in fig. 1 are intended:

1. Switches Q - for switching on and off the circuit in normal and emergency modes.

2. Disconnectors QS - to relieve voltage from de-energized parts of an electrical installation and to create a visible break in the circuit, which is necessary during repair work. Disconnectors, as a rule, are repair and not operational elements.

3. Busbars US - for receiving electricity from sources and distributing it among consumers.

4. Relay protection devices РЗ - for detecting the fact and location of damage in an electrical installation and for issuing a command to disconnect the damaged element.

5. Automation devices A - for automatic switching on or switching of circuits and devices, as well as for automatic regulation of operating modes of electrical installation elements.

6. Measuring devices IP - to control the operation of the main equipment of the power plant and the quality of energy, as well as to account for the generated and supplied electricity.

7. Instrument current transformers TA and TV voltages.

Control questions:

1. Give a definition of the energy system and all the elements included in it.
2. The main parameters of electricity.
3. What energy sources are natural sources?
4. What power plants are called thermal?
5. What are the traditional methods of generating electricity?
6. What methods of generating electricity are non-traditional?
7. List the types of renewable energy sources?
8. List the types of non-renewable energy sources?
9. What types of power plants are thermal power plants?
10. What are the technical and economic advantages of interconnecting energy systems?
11. Which power plants are called basic and which are peak power plants?
12. What are the requirements for energy systems?
13. List the main purposes of automation devices, current and voltage transformers, switches.
14. List the main purposes of disconnectors, relay protection devices and busbars. What is the purpose of a current-limiting reactor?