POWER SYSTEMS – II chapter 1 transmission line modelling.pptx

Transmission line modelling

1. POWER SYSTEMS – II

2. Power Supply Systems Comparison between Various Systems and Copper Efficiencies, Effect of System Voltage on Transmission Efficiency, Choice of Transmission Voltage, Conductor Size and Kelvin’s Law. Transmission Line Constants Transmission line components, Types of conductors, Inductance and Capacitance of Single Phase and Three Phase Lines, Concept of GMDR Mutual GMD Double Circuit Line, Inductance of Composite Conductors, Transposition, Skin Effect and Proximity Effect. Transmission Line Modeling Generalized Network Constants, Modeling of Short Transmission line, Modeling of Medium transmission line: Nominal-T and Nominal-π methods and Long Transmission Lines, Rigorous Line Modeling. Mechanical Design of Transmission Lines Sag and Tension Calculations with equal and unequal heights of towers, effect of Wind and Ice on weight of conductor. Line Supports, Conductor Materials, Overhead Lines Vs Underground Cables. POWER SYSTEMS – II Syllabus

3. Over Head Line Insulators Types of Insulators, String efficiency and Methods for improvement–Numerical Problems, Voltage distribution, Calculation of string efficiency, Capacitance grading and Static Shielding. Under-Ground Cables Types of Cables, Insulation in Cables, Armouring & Covering of Cable, Insulation Resistance OFR Cables, Stress in Insulation, Sheathing in Cable, Use of Inter Sheaths, Capacitance Grading, Capacitance in 3-Core Cables. Corona : Phenomenon of Corona, Critical Voltages, Power Loss due to Corona, Factors Affecting Corona Loss, Radio Interference. Text Books 1. A Text Book on Power Systems Engineering by Sony, Gupta, Bhatnagar and Chakrabarti , Dhanapatrai & Co. 2. Electrical Power Systems by C. L. Wadhwa . Reference Books 1. Electrical Power by S. L. Uppal . 2. A Course in Power Systems by J. B. Gupta. 3. Electrical Power Transmission and Distribution by S. Siva Nagaraju and S. Satyanarayana .

4. Chapter 1 Power Supply Systems Comparison between Various Systems and Copper Efficiencies, Effect of System Voltage on Transmission Efficiency, Choice of Transmission Voltage, Conductor Size and Kelvin’s Law.

5. Transmission System

6. POWER SYSTEM The generation, transmission and distribution of electric power is called power system. A power system has the following stages :- - Generation of electric power. - Transmission of electric power. - Distribution of electric power. Most transmission lines are high-voltage three-phase alternating current (A.C). High-voltage direct-current (HVDC) technology is used for greater efficiency over very long distances. Electricity is transmitted at high voltages (115 kv or above) to reduce the energy loss which occurs in long-distance transmission.

7. LAYOUT OF POWER SYSTEM A power system consists of the following stages :- Power station Primary Transmission Secondary Transmission Primary Distribution Secondary Distribution

8. TRANSMISSION SYSTEM The system by which bulk power is delivered to the load centres and large industrial consumers from power stations is called transmission.

9. Primary Transmission and Secondary Transmission Primary Transmission - High voltages of the order of 66kv, 132kv, 220kv, 400kv are used for transmitting power by 3 phase 3 wire overhead system. This is supplied to substations usually at the outskirts of major distribution centre or city. Secondary Transmission - On the outskirts of the city, there are sub-station which step down the primary transmission voltage to 66kv or 33kv and power is transmitted at this voltage. This forms the secondary transmission system, 3 phase wire system is used.

10. ADVANTAGES OF HIGH VOLTAGE FOR TRANSMISSION Reduces the cost of conductor material. Efficiency of transmission increases. Percentage line drop is reduced.

11. LINE SUPPORTS

12. CONDUCTOR The most commonly used conductor materials for overhead lines are – Copper – This is an ideal conductor material for transmission and distribution of electrical power, but due to high cost and non-availability in abundance, it has limited applications. ( b) Aluminium – This conductor material is next to copper. It is cheaper than copper and is used where straight lines are required, due to non-flexibility. ( c) A.C.S.R ( Aluminium Conductor Steel Reinforced) – These conductors have a central core of galvanised steel whereas aluminium conductors form the outer layer. They are mechanically strong and lighter in weight. Therefore these can be used for longer spans.

13. ..

14. INSULATORS

15. Parameter AC Transmission System DC Transmission System Definition An electric power transmission system that uses alternating current to transmit the power is called the AC transmission system. An electric power transmission system which transmits the electric power in the form of DC supply, is called the DC transmission system. Construction of Transmission Line The construction of AC transmission lines is more complicated. The construction of DC transmission lines is less complicated. Generation In the AC transmission system, electric power can be generated easily at high voltages. In the DC transmission system, electric power cannot easily be generated at high voltage due to commutation problems. Number of Conductors For the AC transmission, three conductors are required, (for red phase, yellow phase and blue phase). For DC transmission, only two conductors (positive and negative) are required. Need of Transformer In the AC transmission, transformer is used for stepping-up or down the voltage. In DC transmission, the transformer cannot be used as the transformer does not work with the DC supply. Insulation Material The AC transmission requires more insulation material due to more number of conductors. The DC transmission requires less insulation material due to less number of conductors. Comparision between AC and DC Transmission Systems

16. Stability and Surges Effects of stability and surges are found in the AC transmission. The DC transmission is free from the effects of stability and surges. Factor Present In AC transmission lines, the capacitance, inductance and phase displacement are present. In DC transmission lines, the capacitance, inductance and phase displacement do not present. Effect of Capacitance In AC transmission lines, the presence of capacitance causes a continuous power loss. In DC transmission lines, no capacitance is present, thus no power loss. Effect of Inductance Inductance is present in the AC transmission lines causes more voltage drop. Therefore, it has poor voltage regulation. Inductance is not present in the DC transmission lines, hence the voltage drop is less. Therefore, it has good voltage regulation. Change in Voltage Level In AC transmission system, the transformer is used for increasing or decreasing the voltage level. In DC transmission system, chopper and booster are used for increasing or reducing the voltage level. Skin Effect The skin effect occurs in the AC transmission system. In case of DC transmission system, the skin effect does not occur. Corona Effect Corona most exists in AC transmission system. Corona rarely exists in DC transmission system.

17. Required Components The main components of AC transmission system are transformer, conductors and insulators. The main components of DC transmission system are rectifier, conductors, insulators and inverter. Cost The construction and maintenance of the AC transmission system is cheap. The construction and maintenance of DC transmission system is expensive. Repair and Maintenance The repair and maintenance of AC transmission system is simple and easy. The repair and maintenance of DC transmission system is difficult as compared to AC transmission system. Length of Conductors The AC transmission system is used for short distance transmission. The DC transmission is used for transmission of electric power for long distances. Interference The AC transmission lines interfere the nearby communication lines. The DC transmission lines do not interfere the communication lines. Dielectric Loss The AC transmission lines have dielectric losses. The DC transmission lines do not have dielectric losses.

18. CLASSIFICATION OF TRANSMISSION LINES We classify transmission lines with reference to :- Voltage Distance A.C or D.C

19. TYPICAL TRANSMISSION VOLTAGE LEVELS the nominal extra high voltage lines in vogue are ±800 kv hvdc & 765kv, 400kv, 230/220 kv , 110kv and 66kv ac lines.

20. DISTANCE Short length T.L (< 80 km) Medium length T.L ( 80-160 km) Long length T.L (>160 km)

21. AC or DC The transmission line may be ac or dc depending upon the application.

22. DC TRANSMISSION ADVANTAGES :- 1. It requires only two conductors as compared to three for a.c transmission. 2. There is no inductance, capacitance, phase displacement and surge problems in d.c. transmission. 3. A d.c. transmission line has better voltage regulation. 4. There is no skin effect in a d.c. system. Therefore, entire cross-section of the line conductor is utilized. 5. For the same working voltage, the potential stress on the insulation is less in case of d.c. system. 6. A d.c. line has less corona loss. 7. The high voltage d.c transmission is free from the dielectric losses. 8. There are no stability problems.

23. DC TRANSMISSION DISADVANTAGES :- 1 . Electric power cannot be generated at high d.c voltage due to commutation problems. 2. The d.c. voltage cannot be stepped up for transmission of power at high voltages. 3. The d.c switches and circuit breakers have their own limitations.

24. AC TRANSMISSION ADVANTAGES :- 1. The power can be generated at high voltages. 2. The maintenance of a.c substations is easier and cheaper. 3. The a.c voltage can be stepped up or stepped down by the transformers with ease and efficiency. This permits to transmit power at high voltages and distribute it at safe potentials.

25. AC TRANSMISSION DISADVANTAGES :- 1. An a.c line requires more copper than a d.c. line. 2. The construction of a.c transmission line is more complicated than a d.c transmission line. Due to skin effect in the a.c system, the effective resistance of the line is increased . 4 . An a.c line has capacitance. Therefore, there is a continuous loss of power due to charging current even when the line is open. 5. An a.c line has corona loss.

26. Various Systems of Power Transmission For the transmission of electrical power from the generating stations to the substations for distribution, there are various types of power transmission systems are adopted. However, for the transmission of electric power, three-phase three wire transmission system is universally adopted. The different possible systems for electric power transmission are discussed below. DC Transmission System When the electrical power is transmitted using direct current or voltage, then the transmission system is called the DC transmission system. The transmission system can be further classified into following three types − DC Two-Wire System DC Two-Wire with Mid-Point Earthed DC Three-Wire System

27. Single Phase AC System When the electrical power is transmitted using alternating current or voltage and in this system any one of the three phases is used for the transmission of electric power. The single phase AC system is also classified into three types viz − Single-Phase Two-Wire System Single-Phase Two-Wire with Mid-Point Earthed System Single-Phase Three-Wire System

28. Two Phase AC System In a two-phase AC system, the electrical power is transmitted in the form of alternating quantities and any two of the three phase are used for transmitting the electric power. Depending upon the conductors used, the two phase AC system is of following two types − Two-Phase Three-Wire System Two-Phase Four-Wire System

29. Three Phase AC System When the electric power is transmitted by using the three phases or three line conductors, then the system is called the three phase AC system. There are two types of three phase AC system as − Three-Phase Three-Wire System Three-Phase Four-Wire System

30. Conductor Material Required in Overhead DC Transmission System Overhead DC Transmission System The overhead transmission system is the one in which the conductors are hanged with the help of pole supports. When the transmission lines carry direct current, then the system is called the DC transmission system. There are three types of overhead DC transmission systems viz. − Two wire DC system with one conductor earthed Two wire DC system with mid-point earthed Three wire DC system

31. Conductor Material Required in Two-Wire DC System with One Conductor Earthed Consider a two wire DC system with one conductor earthed as shown in Figure-1. Here, one is the positive wire and the other is the negative wire and the load is connected between the two wires.

32. Conductor Material Required in Two-Wire DC System with Mid-Point Earthed Consider a two wire DC system with mid-point earthed as shown in Figure-2. In this system, the maximum voltage between any conductor and the earth is 𝑉 𝑚 , therefore the maximum voltage between two conductors is 2𝑉 𝑚 . Therefore, the load current is given by,

33. Conductor Material Required in Three-Wire DC System Consider a three-wire DC system in which two outers and a neutral wire be earthed at the generator end as shown in Figure-3. If a balanced load is connected to the system, no current will flow in the neutral wire. Then, for the balanced load, the load current is given by ,

34. Single-Phase AC Transmission System The electric power transmission system in which two conductors viz.  phase conductor  and  neutral wire  are used to transmit the electric power is known as  single phase AC transmission system . The single phase AC transmission system can be classified into following three types viz. − Single-phase two-wire system with one conductor earthed Single-phase two-wire system with mid-point earthed Single-phase three-wire system Conductor Material Required in Single-Phase Overhead AC Transmission System

35. Conductor Material Required in 1-Phase 2-Wire System with One Conductor Earthed The single phase two wire AC system with one conductor earthed is shown in Figure-1. Let, Power transmitted= P Maximum voltage between conductors= Vm RMS value of voltage= Vm / √2 Power factor of load= cos ϕ Then, the load current is given by,

36. Conductor Material Required in 1-Phase 2-Wire System with Mid-Point Earthed A single phase 2-wire AC transmission system with mid-point earthed is shown in Figure-2. Here, the two wire have equal and opposite maximum voltages ( V m ) with respect to earth. Therefore, the maximum voltage between the two wires is 2V m .

37. Conductor Material Required in 1-Phase 3-Wire System The single-phase three-wire AC system is shown in Figure-3. This system consists of two outer wires and one neutral wire taken from the mid-point of the phase winding. When a balanced load is connected to the system, the current through the neutral wire is zero. Refer the circuit, Maximum voltage between conductors = 2 Vm RMS value of voltage between conductors = 2 Vm/ √2 Power factor of load= cos ϕ

38. Conductor Material Required in Overhead Two-Phase AC Transmission System Two Phase AC Transmission System When the transmission system consists of two line conductors for the transmission of AC electric power from generating station to the substations for its distribution, it is called the  two phase AC transmission system . Depending upon the number of conductors used, the two phase AC transmission system is of two types, viz. − Two Phase Three Wire AC System Two Phase Four Wire AC System

39. Conductor Material Required in Two Phase Three Wire AC System Consider a two-phase three-wire AC system as shown in Figure- There are two line conductors and one neutral wire is taken from the junction of two-phase windings whose voltages are in quadrature (i.e., at 90°) with each other. It is clear that each line conductor transmits one half of the total electric power. Therefore, the RMS value of voltage between any line conductor and the neutral (i.e., phase voltage) wire is,

40. Conductor Material Required in Two Phase Four Wire AC System A two phase four wire AC system is shown in Figure-2. In this system, the four line wires are taken from the ends of the two phase windings and the mid-points of the two phase windings are connected together. Each phase transmits one half of the total power (P). Now, consider any one of the two phases, say phase winding AB, then,

41. Conductor Material Required in Three-Phase Overhead AC Transmission System Three-Phase Transmission System A three-phase transmission system is the one in which three line conductors are used to transmit the AC electric power from generating station to the substations. The three-phase system is universally adopted for transmission of electric power. Depending upon the number of conductors used, the three-phase AC transmission system is classified into two types viz. − Three-Phase Three-Wire System Three-Phase Four-Wire System

42. Conductor Material Required in 3-Phase 3-Wire AC System Consider a three-phase three-wire AC system as shown in Figure-1, it has three line conductors and one earthed neutral wire. The three-phase three-wire system may be star connected (as shown in Figure-1) or delta connected. Let, Maximum voltage per phase,V ph = V m RMS value of voltage per phase= Vm /√2 Power transmitted by per phase= P/3 Therefore, the load current per phase is given by,

43. Conductor Material Required in 3-Phase 4-Wire AC System The three-phase four-wire AC transmission system is shown in Figure-2. In this system, the neutral wire is taken from the neutral point and the cross-sectional are of neutral wire is generally one-half that of the line conductors. If the load connected to the 3-phase 4-wire system is balanced, then current through the neutral wire is zero. Let , Maximum voltage per phase,V ph = V m RMS value of voltage per phase= Vm /√ 2 Power transmitted by per phase= P/3 Also, assume that the load is balanced and power factor of load as cos 𝜙. Then,

44. Underground Electric Transmission System The system of electric power transmission uses underground cables for transmitting the electric power from the generating station to the consumers is known as  underground electric transmission system . Underground cables have low visibility and are not affected by bad weather. However, the cost of these cables is comparatively high and their laying process is also time consuming .

45. Types of Underground Transmission System Based on the type of supply (i.e. AC or DC) and the number of conductors used, the underground electric transmission system can be classified into the following types − Underground DC System Two-Wire DC System Two-Wire DC System with Mid-Point Earthed Three-Wire DC System Underground 1-Phase AC System 1-Phase Two-Wire AC System 1-Phase Two-Wire with Mid-Point Earthed 1-Phase Three-Wire AC System Underground 2-Phase AC System 2-Phase Three-Wire AC System 2-Phase Four-Wire AC System Underground 3-Phase AC System 3-Phase Three-Wire AC System 3-Phase Four-Wire AC System

46. Comparison of Conductor Material in Underground Systems of Transmission In underground system, if the electric power is transmitted using the multi-core belted type cables, then the chief electric stress on the insulation is between conductors. Therefore, under such situations, the comparison of conductor material is made on the basis of maximum voltage between the conductors . For comparing the conductor material needed in the underground electric power transmission system, the following assumptions are to be made − The same amount of electric power (say P watts) is transmitted by each system. The maximum voltage between conductors (say V m ) is the same in each system. The length (say l meters) over which the electric power is transmitted remains the same in each case. The line losses (say W watts) are the same in case of each system.

47. System Volume of Conductor Material (Same maximum voltage between conductors) DC System (i) 2-wire system 1 (ii) 2-wire with mid-point earthed 1 (iii) 3-wire system 1.25 The table given below shows  the ratio of conductor material required in any underground system with respect to the underground 2-wire DC system . Here, cos 𝜙 is the power factor of the load in AC System.

48. What is the Volume of Conductor Material Required in Underground DC System? Underground DC System When the DC electric power is transmitted from the power generating station to the consumer through the underground cables, then the electric power transmission system is called the  underground DC system . The underground DC transmission system is classified into three types − Two-Wire DC System Two-Wire DC System with Mid-Point Earthed Three-Wire DC System

49. The underground two-wire DC system is shown in Figure-1. It has two conductors taken from the generator terminals. Let, 𝑉 𝑚  = Maximum voltage between conductors 𝑃 = Power to be transmitted 𝑙 = Distance for which power is transmitted The load current is given by , Conductor Material Required in Underground 2-Wire DC System

50. Conductor Material Required in Underground 2-Wire DC System with Mid-Point Earthed The circuit diagram of the underground 2-wire DC system with mid-point earthed is shown in Figure-2. Let, 𝑉 𝑚  = Maximum voltage between conductors 𝑃 = Power to be transmitted Then, the load current is given by,

51. Conductor Material Required in Underground Three-Wire DC System The circuit diagram of an underground three-wire DC system is shown in Figure-3. In this system, the two outer line conductors are taken from the outer terminals of the generators and the neutral wire is taken from the mid-point. When the balanced load is connected to the system, the current in the neutral wire will be zero. Let, 𝑉 𝑚  = Maximum voltage between outer conductors 𝑃 = Power to be transmitted Then, the load current in the system is given by,

52. Hence,  the volume of conductor material require in the underground three-wire DC system is 1.25 times that required for the underground 2-wire DC system .

53. Volume of Conductor Material Required in Underground Single-Phase AC System Underground Single-Phase AC System The electric transmission system which transmits the single-phase AC electric power through the underground cables is termed as the  underground single-phase AC system of transmission . The underground single-phase AC system of electric power transmission is classified into three types viz. − Single-Phase Two-Wire AC System Single-Phase Two-Wire AC System with Mid-Point Earthed Single-Phase Three-Wire AC System The various systems of the electric transmission require different volume of conductor material. The volume of conductor material required in the above 1-phase underground AC system is described below.

54. Conductor Material Required in 1-Phase 2-Wire Underground AC System The typical single-phase 2-wire AC underground AC system is shown in Figure-1.

55. Conductor Material Required in Underground 1-Phase 2-Wire AC System with Mid-Point Earthed Figure-2 shows the underground single-phase two-wire AC system with mid-point earthed. Here, Maximum voltage between line wires= Vm RMS value of voltage= Vm / √2 Power factor of the load= cos ϕ Thus, the load current is given by,

56. Conductor Material Required in the 1-Phase 3-Wire Underground AC System The underground single-phase three-wire AC system is shown in Figure-3. In this system, an additional neutral wire is provided. The cross-sectional area of the neutral wire is to be half of the either of the outer wire. Also, if the load connected to the system is balanced. Then, no current flows in the neutral wire. Therefore, the line losses are given by,

57. Volume of Conductor Material Required in Underground Two-Phase AC System When the two-phase AC electrical power is transmitted through the underground cables from the generating station to the consumers, then the transmission system is called the  underground two-phase AC system of transmission . Depending upon the number of conductors used, the underground two-phase AC system is classified into two types viz − Two-phase three-Wire AC System Two-phase four-wire AC System

58. Conductor Material Required in Underground Two-Phase Three-Wire AC System Figure-1 shows the circuit diagram of the underground two-phase three-wire AC system of electric power transmission . Let us consider that the maximum voltage between the outer conductors is  Vm . Then, the maximum voltage between any one outer wire and neutral wire is  Vm /√ 2 .  It is because the voltages in the two phase windings are 90° out of phase. Therefore, the RMS value of voltage between outer wire and the neutral wire is given by , RMS voltage between outer and neutral wire= Vm /√ 2/√2= Vm /2 And if P is the power to be transmitted. Then, each conductor carries one half of the total power. Hence, the load current in each outer wire is,

59. Conductor Material Required in Underground Two-Phase Four-Wire AC System The underground two-phase three-wire AC system of electric power transmission is shown in Figure-2. This system can be considered as two independent 1-phase systems, each transmitting one-half of the total power . Let, Maximum voltage between outer two conductors= Vm ∴ RMS value of voltage= Vm /√2 Power to be transmitted=P Therefore, the load current in the outer line conductors is given by,

60. Underground Three Phase AC System When the three phase AC electric power is transmitted through the underground cables, then the system of electric power transmission is called the  underground three phase AC system . Depending upon the number of conductors used, the underground three-phase AC system of electric power transmission is classified into two types − Three-Phase Three-Wire System Three-Phase Four-Wire System

61. Conductor Material Required in Underground Three-Phase Three-Wire System The circuit diagram of the underground 3-phase 3-wire AC system is shown in Figure-1. In this system, the three line conductors are taken from the three phase windings of the alternator . Let the maximum value of voltage between line conductors is  V m . Then, per phase maximum voltage is 𝑉 𝑚 /√3. Therefore, the RMS value of the per phase voltage is given by,

62. Conductor Material Required in Underground Three-Phase Four-Wire System The underground 3-phase 4-wire AC system for electric power transmission is shown in Figure-2. There are three line conductors taken from the outer terminals of the windings of alternator and the fourth one is the neutral wire taken from the common terminal of three windings. If the load connected to the system is balanced, then no current flows through the neutral wire. As a result, the four system reduces to 3-wire system. Let , Vm =Maximum voltage between line conductors Vm /√2=Maximum voltage per phase P/3=Power transmitted per phase

63. Assuming the area of cross section of neutral wire to be half that of line conductor. Therefore, the volume of conductor material required in the underground 3-phase 4-wire AC system, say K 1 , is given by,

64. Transmission System Same Maximum Voltage to Earth Same Maximum Voltage between Conductors                                                      DC Transmission System Two-wire system                      1                      1                   Two-wire with mid-point earthed                    0.25                       1       Three-wire system                  0.3125                    1.25 The following table gives the ratio of conductor material in any system with respect to that in the corresponding two-wire DC system −

65. From the above comparison table, it is clear that − There is a great saving in conductor material if the DC system is adopted for the transmitting the electric power. In case of AC system, the three-phase AC system is most suitable system for transmitting the power due to saving in the conductor material. Also, this system is convenient and efficient than the other systems.

66. Economic Choice of Conductor Size – Kelvin’s Law (Economics of Power Transmission) The cost of conductor material required for designing a transmission line is a very considerable part of the total cost of a transmission line. Therefore, the determination of proper size of the conductor for the transmission line is very important. The proper size of the conductor for transmission line is determined by the  Kelvin’s law  (given by Lord Kelvin in 1881). The Kelvin’s law states that the most economical area of conductor is that for which the total annual cost of transmission line is minimum. The total annual cost of the transmission line can be divided into two parts viz. − Annual charges on capital cost Annual cost of energy wasted in conductor

67. Annual Charges on Capital Cost These annual charges are on the account of interest and depreciation on the capital cost of complete installation of the transmission line. In case of overhead transmission system, the complete installation cost will be the annual interest and depreciation on the capital cost of conductors, pole supports and insulators and also the cost of their erection. Now, for the overhead transmission system, the cost of conductor is proportional to the area of cross-section, the cost of insulator is constant and the cost of pole (or tower) supports and their erection is partly constant and partly proportional to cross-sectional area of the conductor. Therefore, the annual charges on the capital cost of an overhead transmission line can be expressed as − Annual Charges=P1+aP2 ⋅⋅⋅(1) Where, P 1  and P 2  are the constants and a is the cross-sectional area of the conductor.

68. Annual Cost of Energy Wasted in Conductor This cost is on the account of energy wasted in the conductor due to I 2 R losses. Suppose a constant current in the conductor throughout the year, thus the power loss in the conductor is directly proportional to the resistance. Since, the resistance of the conductor is inversely proportional to the cross-sectional area of the conductor. Hence, the energy lost in the conductor is inversely proportional to the cross-sectional area. Therefore, the annual cost of energy wasted in the conductor of overhead transmission line can be expressed as −

69. i.e., Variable part of annual charges on capital cost=Annual cost of energy wasted Hence,  the  Kelvin’s law  can also be stated as the most economical area of conductor is that for which the variable part of annual charges on capital cost is equal to the annual cost of energy wasted in the conductor. The figure shows the graphical representation of the Kelvin’s law. Here, m is the lowest point on the curve, which represents the most economical area of cross-section, i.e., lowest annual cost of transmission line.

70. Limitations of Kelvin’s Law In practice, the following are the limitations of the Kelvin’s law − The Kelvin’s law does not involve several physical factors such as safe current density, mechanical strength, corona loss, etc. The conductor size determined by the Kelvin’s law may not always be practicable one since it may be too small for the safe carrying of necessary current. It is not easy to estimate the energy loss in the transmission line without actual load curves, which are not available at the time of estimation. The assumption that the annual charges on the account of interest and depreciation on the capital cost is in the form of 𝑃 1  + 𝑎𝑃 2  is strictly speaking not true. Therefore, the interest and depreciation on the capital cost cannot be determined accurately.