Power systems
Long Transmission Line :
A transmission line having a length more than 240 km is consider as a long transmission line. In a long transmission line, parameters are uniformly distributed along the whole length of the line. For a long transmission line, it is considered that the line may be divided into various sections, and each section consists of an inductance, capacitance, resistance and conductance as shown below.
Let’s consider a bit smaller part of a long transmission line having length ‘ds’ situated at a distance ‘s’ from the receiving end. Series impedance of the line is represented by ‘zds’ and ‘yds’ is the shunt impedance of the line. Due to charging current and corona loss the current is not uniform along the line. Voltage is also different in different parts of the line because of inductive reactance.
Where,
- r-resistance per unit length, per phase
- l–inductance per unit length, per phase
- c-capacitance per unit length, per phase
- x–inductive reactance per unit length, per phase
- z–series impedance per unit length, per phase
- g–shunt leakage conductance, per phase to neutral per unit length
- b – shunt leakage susceptance, per phase to neutral per unit length
- y – shunt admittance per unit length, per phase to neutral.
For constant supply let,
- V–voltage at a distance ‘s’ from the load end
- V + dV-voltage at a distance (s+ds) from the load end
- I–current at a distance ‘s’ from the load end
- I + dI–current at a distance (s+ds) from the load end.
The difference in the voltage between the ends of the assumed sections of length ds is dV. This difference is caused by the series impedance of the line.
Similarly, the difference between the two ends of the section resulting from the shunt admittance of the line is given by the equation
for knowing the value of V, differentiate equation (1) with respect to ‘s’,
and for current differentiate equation (2)
equation (3) and (4) are similar in form and therefore their general equations are also similar.
The equation (5) is the linear differential equation with constant coefficients. The general solution of this equations is
Where, C1 and C2 are the arbitrary constants, and it is found from the known value of the V and I at some point of the line. For determining the value of I differentiate the above equation with respect to ‘s’
on combining the above equation with equation (1) we get,
substituting the value of ϒ = √zy in equation (7) gives
The value of V and I at the receiving end where s = 0, is given by the equations
The values of C1 and C2 are found from the simultaneous equations shown below
and
The value of C1 and C2 are substituted in general equations of voltage and current to obtain the steady-state values of V and I at any intermediate point distant ‘s’ from the receiving end.
and
For governing the behaviour of transmission line in steady-state equation (13) and (14) are used. These equations can also be written in hyperbolic form by using hyperbolic constant shown below
Substitutes the hyperbolic constant in equations (13) and (14) gives
these equations can also be written as sending end voltage and current equations by replacing s = S
The ABCD parameters are defined below
These equations help in evaluating the performance of the long line.
Power systems
Surge Impedance Loading :
Capacitance and reactance are the main parameters of the transmission line. It is distributed uniformly along the line. These parameters are also called distributed parameters. When the voltage drops occur in transmission line due to inductance, it is compensated by the capacitance of the transmission line.
The transmission line generates capacitive reactive volt-amperes in its shunt capacitance and absorbing reactive volt-amperes in its series inductance.The load at which the inductive and capacitive reactive volt-amperes are equal and opposite, such load is called surge impedance load.
It is also called natural load of the transmission line because power is not dissipated in transmission. In surge impedance loading, the voltage and current are in the same phase at all the point of the line. When the surge impedance of the line has terminated the power delivered by it is called surge impedance loading.
Shunt capacitance charges the transmission line when the circuit breaker at the sending end of the line is close. As shown below
Let
- V = phase voltage at the receiving end
- L = series inductance per phase
- XL = series inductance reactance per phase
- XC = shunt capacitance reactance per phase
- Zo = surge impedance loading per phase
Capacitive volt-amperes (VAr) generated in the line
The series inductance of the line consumes the electrical energy when the sending and receiving end terminals are closed.
Inductive reactive volt-amperes (VAr) absorbed by the line
Under natural load, the reactive power becomes terminated, and the load becomes purely resistive.
And it is calculated by the formula given below
Surge impedance loading is also defined as the power load in which the total reactive power of the lines becomes zero. The reactive power generated by the shunt capacitance is consumed by the series inductance of the line.
If Po is its natural load of the lines, (SIL)1∅ of the line per phase
Since the load is purely resistive,
Thus, per phase power transmitted under surge impedance loading is (VP2)/ZO watts, Where Vp is the phase voltage.
If kVL is the receiving end voltage in kV, then
Surge impedance loading depends on the voltage of the transmission line. Practically surge impedance loading always less than the maximum loading capacity of the line.
If the load is less than the SIL, reactive volt-amperes are generated, and the voltage at the receiving end is greater than the sending end voltage. On the other hand, if the SIL is greater than the load, the voltage at receiving end is smaller because the line absorbs reactive power.
If the shunt conductance and resistance are neglected and SIL is equal to the load than the voltage at both the ends will be equal.
Conclusion :
Surge impedance load is the ideal load because the current and voltage are uniform along the line. The wave of current and voltage is also in phase because the reactive power consumed are equal to the reactive power generated by the transmission line.
Power systems
Definition of Corona Effect :
This phenomenon of ionization of surrounding air around the conductor due to which luminous glow with hissing noise is rise is called corona effect.
For overhead transmission system the atmospheric air, which is the dielectric medium, behaves practically like a perfect insulator when the potential difference between the conductor is small. If the voltage impressed between the conductor is of alternating nature, the charging current will flow due to the capacitance of the lines. This charging current increases the voltages of the lines and corresponding increase the electric field intensity of the lines.
When the value of electric field intensity is less than 30kV (disruptive voltage), the flow of current between two conductors of the lines is negligibly small. But when the electric field intensity reaches this critical value or disruptive voltage the airs between the conductors get ionizes and becomes conducting. If the voltage goes on increasing spark is established between the conductors until the complete breakdown of the insulating properties of the material.
Contents:
- Disadvantages of corona discharge
Corona Formation:
Air is not perfect insulator and even under normal condition, the air contains many free electrons and ions. When an electric field intensity is established between the conductors, these ions and free electrons experience force upon them. Due to this force, the ions and free electrons get accelerated and moved in the opposite direction.
The charged particles during their motion collide with one another and also with the very slow moving uncharged molecules. Thus, the number of charge particles goes on increasing rapidly. This increase the conduction of air between the conductors and a breakdown occurs. Which established the arc or discharge between the conductors.
Factors affecting corona:
Corona loss depends on a large number of factors, the most important being broadly classified in the following way:
Effect of supply voltage : If the supply voltage is high corona loss is higher in the lines. In low voltage transmission lines the corona is negligible, due to the insufficient electric field to maintain ionization.
The condition of conductor surface : If the conductor is smooth, the electric field will be more uniform as compared to the rough surface. The roughness of conductor is caused by the deposition of dirt, dust and by scratching, etc. Thus, rough line decreases the corona loss in the transmission lines.
Air Density Factor : The corona loss in inversely proportional to air density factor, i.e., corona loss, increase with the decrease in density of air. Transmission lines passing through a hilly area may have higher corona loss than that of similar transmission lines in the plains because in a hilly area the density of air is low.
Effect of system voltage : Electric field intensity in the space around the conductors depends on the potential difference between the conductors. If the potential difference is high, electric field intensity is also very high and hence corona is also high. Corona loss, increase with the increase in the voltage.
A spacing between conductors : If the distance between two conductors is much more as compared to the diameter of the conductor than the corona loss occurs in the conductor. If the distance between them is extended beyond certain limits then the dielectric medium between them get decreases and hence the corona loss also reduces.
Disadvantages of corona discharge:
- A faint glow appears around the conductor which is visible in the dark.
- There is the tendency in the conductor to vibrate.
- Corrosion due to ozone and oxides of nitride formation.
- There is the loss of power.
- There is radio and television interference
- Harmonic current flows resulting from corona formation.
Minimizing corona:
Corona decreases the efficiency of transmission lines. Therefore, it is necessary to minimize corona.The following factors may be considered to control corona:
Conductor diameter : For reducing corona loss this method of increasing conductor diameters is very effective. Diameters of conductors can be increased by using hollow conductors and by using steel-cored aluminum conductors(ACSR) conductors.
The voltage of the line : Voltage of transmission lines is fixed by economic considerations. To increase the disruptive voltage the spacing of the conductors is to be increased, but this method has some limitations.
Spacing between conductors : If the space between conductors increases, then the voltage drops between them also increases due to increase in inductive reactance.
Important points:
- Disruptive voltage is the minimum voltage at which the breakdown of air occurs and corona starts.
- Visual critical voltage is the minimum voltage at which visible corona begins.
Power systems
Corona Power Loss :
In the previous topic, we learn about the corona effect due to which several losses occurs in transmission lines. These losses decrease the efficiency of transmission lines. Out of all the losses the corona power loss is the one which affects most, the proficiency of lines.
The power dissipated in the system due to corona discharges is called corona loss. Accurate estimation of corona loss is difficult because of its variable nature. It has been found that the corona loss under fair weather condition is less than under foul weather conditions. The corona loss under appropriate weather conditions is given below by the Peek’s formula;
Where,
- f – frequency of supply in Hz
- En – r.m.s phase voltage in kV
- Eo – disruptive critical voltage per phase in kV
- r – radius of the conductor in meters
- D – spacing between conductors in meters
It is also to be noticed that for a single –phase line,
En=1/2×line voltage
and for a three phase line,
En = 1/(√3)×line voltage
Peek’s formula is applicable for decided visual corona. This formula the gives the inaccurate result when the losses are low and En/Eo is less than 1.8. It is superseded by Peterson’s formula given below;
Where,
- f – frequency of supply in Hz
- r – radius of the conductor
- D – spacing between conductors in meters
Factor F is called the corona loss function. It varies with the ratio (En/Eo). Eo is calculated by the formula given below,
Where,
- Go= maximum value of disruptive critical voltage gradient in V/m.
Power systems
Transmission Lines :
A transmission line is used for the transmission of electrical power from generating substation to the various distribution units. It transmits the wave of voltage and current from one end to another. The transmission line is made up of a conductor having a uniform cross-section along the line. Air act as an insulating or dielectric medium between the conductors.
For safety purpose, the distance between the line and ground is much more. The electrical tower is used for supporting the conductors of the transmission line.Tower are made up of steel for providing high strength to the conductor. For transmitting high voltage, over long distance high voltage direct current is used in the transmission line.
Parameters of transmission line :
The performance of transmission line depends on the parameters of the line. The transmission line has mainly four parameters, resistance, inductance, capacitance and shunt conductance. These parameters are uniformly distributed along the line. Hence, it is also called the distributed parameter of the transmission line.
The inductance and resistance form series impedance whereas the capacitance and conductance form the shunt admittance. Some critical parameters of transmission line are explained below in detail
1. Line inductance : The current flow in the transmission line induces the magnetic flux.When the current in the transmission line changes, the magnetic flux also varies due to which emf induces in the circuit. The magnitude of inducing emf depends on the rate of change of flux. Emf produces in the transmission line resist the flow of current in the conductor, and this parameter is known as the inductance of the line.
2. Line capacitance : In the transmission lines, air acts as a dielectric medium. This dielectric medium constitutes the capacitor between the conductors, which store the electrical energy, or increase the capacitance of the line. The capacitance of the conductor is defined as the present of charge per unit of potential difference.
Capacitance is negligible in short transmission lines whereas in long transmission; it is the most important parameter. It affects the efficiency, voltage regulation, power factor and stability of the system.
3. Shunt conductance : Air act as a dielectric medium between the conductors. When the alternating voltage applies in a conductor, some current flow in the dielectric medium because of dielectric imperfections. Such current is called leakage current. Leakage current depends on the atmospheric condition and pollution like moisture and surface deposits.
Shunt conductance is defined as the flow of leakage current between the conductors. It is distributed uniformly along the whole length of the line. The symbol Y represented it, and it is measured in Siemens.
Performance of transmission lines :
The term performance includes the calculation of sending end voltage, sending end current, sending end power factor, power loss in the lines, efficiency of transmission, regulation and limits of power flows during steady state and transient conditions. Performance calculations are helpful in system planning. Some critical parameters are explained below
Voltage regulation : Voltage regulation is defined as the change in the magnitude of the voltage between the sending and receiving ends of the transmission line.
The efficiency of transmission lines : Efficiency of the transmission lines is defined as the ratio of the input power to the output power.
Important points of Transmission lines :
- Admittance measures the capability of an electrical circuit or we can say it measures the efficiency of a transmission line, to allows AC to flow through them without any obstruction. It SI unit is Siemens and denoted by the symbol Y.
- Impedance is the inverse of the admittance. Its measure the difficulty occurs in the transmission line when the AC flow. It is measured in ohms and represented by the symbol z.
Power systems
Definition of Charging Current in Transmission Line :
In a transmission line, air acts as a dielectric medium between the conductors. When the voltage is applied across the sending end of the transmission line, current starts flowing between the conductors (due to imperfections of the dielectric medium). This current is called the charging current in the transmission line.
In other words, we can say, the current associated with the capacitance of a line is known as the charging current.The strength of the charging current depends on the voltage, frequency, and capacitance of the line. It is given by the equations shown below.
For a single-phase line, the charging current
Where,
C= line-to-line in farads
Xc= capacitive reactance in ohms
V= line voltage in volts
Also, reactive volt-ampere generated by the line = charging volt-amperes of the lines
For a three phase line, the charging current phase
where ,
Vn =voltage to neutral in volts = phase voltages in volts
Cn = capacitance to neutral in farads
Reactive volt-ampere generated by the line = charging volt-amperes of the lines
where ,
Vt = line-to-line voltage in volts.
Significance of charging current :
- It reduces the load current, due to which line losses decreases, and hence the efficiency of the line is increased.
- It improves the power factor of the transmission line.
- Charging current improves the load capacity of the line.
- It improves the voltage regulation of the line because the voltage drop is quite small.
Power systems
Gauss Seidel Method :
Gauss-Seidel Method is used to solve the linear system Equations. This method is named after the German Scientist Carl Friedrich Gauss and Philipp Ludwig Siedel. It is a method of iteration for solving n linear equation with the unknown variables. This method is very simple and uses in digital computers for computing.
The Gauss-Seidel method is the modification of the gauss-iteration method.This modification reduces the number of iteration. In this methods the value of unknown immediately reduces the number of iterations, the calculated value replace the earlier value only at the end of the iteration. .Because of it, the gauss-seidel methods converges much faster than the Gauss methods. In gauss seidel methods the number of iteration method requires obtaining the solution is much less as compared to Gauss method.
Let us understand the Gauss-Seidel Method with the help of an example. Consider the total current entering the kth bus of an ‘n’ bus system is given by the equation shown below.
The complex power injected into the kth bus is given as
The complex conjugate of the above equation becomes
Elimination of Ik from the equation (1) and (4) gives
Therefore, the voltage at any bus ‘k’ where Pk and Qk are specified is given by the equation shown below.
Equation (6) shown above is the major part of the iterative algorithm.
At the bus 2, the equation becomes
At the bus 3, the equation becomes
Now for the kth bus, the voltage at the (r + 1)th iteration is given by the equation shown below.
In the above equation, the quantities Pk, Qk, Ykk and Yki are known, and they do not vary during the iteration cycle.
Now the value of Ck and Dk are shown below, which is computed in the beginning, and it is used in every iteration step.
For the kth bus, the voltage at the (r + 1) th iteration can be written as shown below.
Acceleration Factors in Gauss-Seidel Method:
In the Gauss-Seidel method, a large number of the iteration is required to arrive at the specified convergence. The rate of convergence can be increased by the use of the acceleration factor to the solution obtained after each iteration. The Acceleration factor is a multiplier that enhances correction between the values of voltage in two successive iterations.
Let us consider the acceleration Factor for the ith bus.
- Vi(r) is the value of the voltage at the rth iteration.
- Vi(r + 1) is the value of the voltage at the (r + 1)th iteration.
- Vi( accelerated)(r + 1) is the accelerated new value of the voltage at the (r+ 1) th iteration.
- α is the accelerating factor
Then,
Thus, after calculating Vi(r + 1) at ( r + 1)th iteration, we calculate the value of new estimated bus voltage Vi( accelerated)(r + 1) and this new value replaces the previously calculated value. For real and imaginary components of the voltage different accelerating factors are used.
If Vi is resolved into real and imaginary components as
If α and β are the acceleration factor associated with ai and bi then the equation becomes as shown below.
The choice of a specific value of the acceleration factor depends upon the system parameters. The optimum value of α usually lies in the range of 1.2 to 1.6 for most of the systems.
and
