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A modern and civilized society is so much dependent on the use of electrical energy because it has been the most powerful vehicle for facilitating economic, industrial, and social developments. Electrical energy produced at power stations is transmitted to load centres from where it is distributed to its consumers through the use of transmission lines run from one place to another. As a result of the physical properties of the transmission medium, some of the transmitted power is lost to the surroundings. The overall effect of power losses on the system is a reduction in the quantity of power available to the consumers. An accurate knowledge of transmission losses is hinged on the ability to correctly predict the available current and voltage along transmission lines. Therefore, mathematical physics expressions depicting the evolution of current and voltage on a typical transmission line were formulated, and derived therefrom were models to predict available current and voltage, respectively, at any point on the transmission line. The predictive models evolved as explicit expressions of the space variable and they are in close agreement with empirical data and reality.

The importance of electric power in today's world cannot be overemphasized for it is the key energy source for industrial, commercial, and domestic activities [

Electrical energy is generated at power stations which are usually situated far away from load centers. As such, an extensive network of conductors between the power stations and the consumers is required. This network of conductors may be divided into two main components, called the transmission system and the distribution system. The transmission system is to deliver bulk power from power stations to load centers and large industrial consumers while the distribution system is to deliver power from substations to various consumers.

The efficiency of the transmission component of the electric power system is known to be hampered by a number of problems, especially in third-world countries. The major problems identified in [

From the physics of electric power transmission, when a conductor is subjected to electric power (or voltage), electric current flows in the medium. Resistance to the flow produces heat (thermal energy) which is dissipated to the surroundings. This power loss is referred to as ohmic loss [

One way of mitigating losses in the process of transmitting electric power is to apply some strategies to reduce the losses. Ramesh et al. [

In this paper, we propose mathematical models for predicting available current and voltage as well as the power losses along a typical transmission line so as to be able to reckon the net electric power available to be used to meet customers’ demands. In the process, the evolution of current and voltage on the transmission line is studied and models to predict both current and voltage were constructed. In the end, the desired model for predicting power losses along transmission lines were formulated by reframing the power loss function as a mathematical physics problem. This strategy led to the exclusion of all the transmission parameters from the model.

In the next section, we derive the equations that characterise the evolution of electric current and voltage on a typical transmission line. In Section

In this section, we derive the expressions which voltage and current must satisfy on uniform transmission lines. A real transmission line will have some series resistance associated with power losses in the conductor [

Herein, we are interested in determining the extent to which voltage and current outputs differ from their input values over an elemental portion of the transmission line. As such, we consider an equivalent circuit of a transmission line of length

Equivalent circuit of a transmission line.

Solving power (voltage) flow equation (

With the aid of the last two equations, the quantity of current and voltage at any point on the transmission line can be discerned. Table

Current and voltage along a 330 kV single circuit of a typical transmission network with

Length of line (km) | Current (A) | Voltage (kV) |
---|---|---|

10 | 19.14 | 329.5 |

20 | 19.09 | 329.1 |

50 | 19.00 | 327.6 |

100 | 18.87 | 325.3 |

200 | 18.60 | 320.7 |

300 | 18.33 | 316.1 |

In the next section, we seek to predict the total power losses over a typical transmission line.

The main reason for losses on transmission lines is the resistance of the conductor against the flow of current [

The formation of corona on transmission lines is associated with loss of power too, which will have some effect on the efficiency of the transmission line [

The total losses on a transmission line is then given as

It would have sufficed to substitute

Solving (

Table

Predicted power losses for a 330 kV single circuit of the typical transmission network.

Length of line in km | Power losses (in MW) for a load of 100 MW |
Power losses (in MW) for a load of 200 MW |
Power losses (in MW) for a load of 300 MW |
---|---|---|---|

10 | 0.0500 | 0.1800 | 0.4300 |

20 | 0.1000 | 0.3598 | 0.8594 |

50 | 0.2500 | 0.8983 | 2.1438 |

100 | 0.5000 | 1.7927 | 4.2724 |

200 | 0.9953 | 3.5694 | 8.4839 |

300 | 1.4892 | 5.3301 | 12.6354 |

Table

Simulated results of power losses on 330 kV single circuit of the Nigerian transmission network.

Length of line in km | Power losses (in MW) for a load of 100 MW | Power losses (in MW) for a load of 200 MW | Power losses (in MW) for a load of 300 MW |
---|---|---|---|

10 | 0.05 | 0.18 | 0.43 |

20 | 0.09 | 0.37 | 0.87 |

40 | 0.18 | 0.73 | 1.75 |

60 | 0.26 | 1.10 | 2.84 |

100 | 0.41 | 1.85 | 4.66 |

200 | 0.76 | 3.77 | 10.86 |

300 | 1.10 | 5.85 | 24.40 |

Perusal of the results presented in Tables

The evolution of current and voltage on high tension transmission lines as well as the power losses when modelled evolved as second order ordinary differential equations. With appropriate boundary conditions, the solutions obtained are prescribed in closed forms. Allotting values to the input factors, numerical values were obtained for the requisite factors—current, voltage, and power losses.

An advantage of the analytic expressions obtained in this study is that numerical values can be computed with a hand-held calculator unlike in the method of Král et al. [

Based on the above observations, the models can be used to predict requisite electrical measures along typical transmission lines. With these measures, electric transmission-related activities can be planned with a view to enhancing efficiency of the electric power system.

The equations describing the evolution of current and voltage along transmission lines have been utilised to fashion tools to predict requisite electrical measures such as current, voltage, and power losses.

The evolution of current and voltage on transmission lines is a process that can aid the determination of current and voltage as a function of the space variable

The authors declare that there is no conflict of interests regarding the publication of this paper.