# To Develop Mathematical Models For Power Losses Along Transmission Lines And To Minimize The Losses Using Classical Optimization Techniques

## Abstract

Availability of electric power has been the most powerful vehicle for facilitating economic, industrial and social developments of any nation. Electric power is transmitted by means of transmission lines which deliver bulk power from generating stations to load centres and consumers. For electric power to get to the final consumers in proper form and quality, losses along the lines must be reduced to the barest minimum. A lot of research has been carried out on analysis and computation of losses on transmission lines using reliability indices, but hardly any on the minimization of losses using analytical methods. In another vein, a large body of literature exists for the solution of optimal power flow problems using evolutionary methods, but none of them has employed the versatile tool of mathematical modelling.

Thus, the goal of this work is to use the classical optimization approach coupled with the mathematical modelling technique to minimize the transmission power losses.

Specifically, the objectives of the study were to:

- Develop mathematical models for power flow and power losses along electric power transmission lines and solve the mathematical models for electric power flow along transmission lines using an analytical method;
- Develop empirical models of power losses as functions of distance; and
- Minimize the power losses using the classical optimization technique.

In the research, I employed Kirchoff ’s circuit laws and a combination xiii of corona and ohmic losses in obtaining the mathematical models for the power flow and power losses respectively. Empirical models of the power losses were developed using regression analysis.

The findings of this study were:

- The models for power flow along transmission lines evolved as homogeneous second-order partial differential equations which were solved analytically using the method of Laplace transform;
- The model for power losses over the transmission lines was obtained as the sum of the ohmic and corona losses;
- The empirical models developed are monotonic increasing functions of distance. Thus, establishing that power losses increases with distance;
- Power losses are minimized when the operating transmission voltage is equal to the critical disruptive voltage.

With the above results, a workable strategy can be formulated to reduce to the barest minimum electric power losses along transmission lines so as to ensure availability of electric power, in proper form and quality, to consumers.

Hence, this research work has addressed the problem of minimizing electric power losses during transmission.

## Chapter One

### General Introduction

#### 1.1 Background To The Study

Energy is a basic necessity for the economic development of a nation. There are different forms of energy, but the most important form is the electrical energy, (Gupta, 2008) A modern and civilized society is so much dependent on the use of electrical energy. Activities relating to the generation, transmission and distribution of electrical energy have to be given the highest priority in the national planning process of any nation because of the importance of electrical energy to the economic and social development of the society. In fact, the greater the per capital consumption of electrical energy in a country, the higher the standard of living of its people. Therefore, the advancement of a country is measured in terms of its per capital consumption of electrical energy, (Gupta, 2008).

Power plants’ planning in a way to meet the power network load demand is one of the most important and essential issues in power systems. Since transmission lines connect generating plants and substations in power network, the analysis, computation and reduction of transmission losses in these power networks are of great concern to scientists and engineers.

A lot of research works have been carried out on the above listed aspects.(Zakariya, 2010) made a comparison between the corona power loss associated with HVDC transmission lines and the ohmic power loss. The corona power loss and ohmic power loss were measured and computed for different transmission line configurations and under fair weather and rainy conditions. It was pointed out in the work that the general trend of neglecting the corona power loss is not always valid.

It was found from the comparison that, when the transmission line is moderately or lightly loaded, the percentage of corona power loss to ohmic power loss could reach up to one hundred percent especially if the transmission line is operating at a voltage well above the corona onset value. This percentage is also found to increase substantially under rainy conditions. Finally, it was also discovered that, the ratio of corona to ohmic power loss, decreases with increasing number of bundles. (Numphetch et al., 2011) worked on loss minimization using optimal power flow based on swarm intelligences.

(Thabendra et al., 2009) considered multi-objective optimization methods for power loss minimization and voltage stability while (Abdullah et al., 2010) looked at transmission loss minimization and power installation cost using evolutionary computation for improvement of voltage stability.(Bagriyanik et al., 2003) used a fuzzy multi-objective optimization and genetic algorithm-based method to find optimum power system operating conditions. In addition to active power losses, series reactive power losses of transmission system were also considered as one of the multiple objectives.

(Onohaebi, 2010) considered the relationship between distance and loadings on power losses using the existing 330 KV Nigerian transmission network as a case study in his empirical modelling of power losses as a function of line loadings and lengths in the Nigeria 330 KV transmission lines while (Moghadam, 2010) developed a new method for calculating transmission power losses based on exact modelling of ohmic loss. (Ramesh et al., 2009) looked at minimization of power loss in distribution networks by using feeder restructuring, implementation of distributed generation and capacitor placement method. (Lo, 2006)considered feeder reconfiguration for losses reduction in distribution systems. Others who researched into power losses include(Rugthaicharoencheep, 2009),

(Crombie, 2006), to mention a few, various researchers have also worked on the flow of power on electrical networks. (Pandya, 2008) presents a comprehensive survey of various optimization methods for solving optimal power flow problems. The methods considered in the work include linear programming, Newton-Raphson, quadratic programming, nonlinear programming, interior point and artificial intelligence. Under the artificial intelligence method, the following were also considered artificial neural network method, fuzzy logic method, genetic algorithm method, evolutionary programming method, ant colony optimization method and particle swarm optimization method.

It was found in the paper that the classical methods have a lot of limitations. In most cases, mathematical formulations have to be simplified to get the solutions because of the extremely limited capability to solve real-world large-scale power system problems. The classical methods are weak in handling qualitative constraints and they have very poor convergence. The methods are also very slow and computationally expensive in handling large-scale optimal power flow problems. It was also discovered in the paper that the artificial intelligence methods are relatively versatile for handling various qualitative constraints and that the methods can find multiple optimal solutions in a single simulation.

They are therefore suitable in solving multi-objective optimization problems.(William, 2002) looked at alternative optimal power flow formulations while(Claudio et al., 2001) worked on comparison of voltage security constraint using optimal power flow techniques.

(Roya et al., 2008) considered power flow modelling for power systems with dynamic flow controller. Other researchers who also worked on power flow include (Bouktir et al., 2004).

In addition, several researchers have also worked on electric power systems. (Aderinto, 2011) worked on an optimal control model of the electric power generating system. In the research work, she developed a mathematical model for the electric power generating system using the optimal control approach and characterized the mathematical model by prescribing the conditions for the optimality of the electric power generating system and the analytic requirements for the existence and uniqueness of the solution to the system. The optimality condition for the model was determined and the model was solved analytically and numerically. In the study, two control variables were identified, the first for load shedding among the generators in the system and the second for restriction on the capacity of the generators.

The problem was formulated based on the second control variable since the first control variable can only be on or off as the case may be. The optimality conditions for the system were imposed implicitly on the controls and the mathematical model represents a stable loss-free generating system. From the work, it was shown that the generation loss can be controlled and stabilized. (Oke et al., 2007) considered the perspectives on electricity supply and demand in Nigeria while (Ibe, 2007)looked at optimized electricity generation in Nigeria.

(Bamigbola, 2009)characterized an optimal control model of electric power generating system.(Karamitsos, 2006) considered an analysis of blackout for electric power transmission systems while (Aderinto et al., 2010) looked at optimal control of air pollution with application to power generating system model. Others whose researches touched on electric power systems include (Savenkov, 2008) to list a few. As such, much emphasis has been on proper design of electrical power systems and reduction of losses using feeder reconfiguration and evolutionary techniques.

Loss minimization is a critical component for efficient electric power supply systems.

Losses in an electric power system should be around 3 percent to 6 percent,(Ramesh et al., 2009). In developed countries, it is not greater than 10 percent. However, in developing countries it is still over 20 percent, (Ramesh et al., 2009). Therefore stakeholders in the power sector are currently interested in reducing the losses on electric power lines to a desired and economic level. The purpose of this research work, therefore, is to develop mathematical models for power losses along transmission lines and to minimize the losses using classical optimization techniques.

#### 1.2 Objectives Of The Study

Power losses result in lower power availability to the consumers, leading to inadequate power to operate their appliances. High efficiency of power system is determined by its low power losses. The goal of this research work therefore is to use classical optimization techniques to minimize the transmission power losses on transmission lines. The objectives of the research work are to:

- Develop mathematical models for electric power flow and power losses along electric power transmission lines;
- Solve the mathematical models for electric power flow along transmission lines analytically;
- Develop empirical models of power losses as functions of distance;
- Minimize power losses using the classical optimization technique.

#### 1.3 Significance Of The Study

The mathematical representation of power flow along transmission lines provides a better understanding of the flow of electric power on transmission lines and the evolution of voltage and current along the lines. The mathematical representation of power losses along transmission lines gives an insight into the major problems on electric power transmission.

The minimization of losses on electric power transmission lines using classical optimization technique provides a solution, in a compact form, to the major problem encountered in power transmission.

#### 1.4 Organization Of The Thesis

This dissertation report is divided into five chapters. The first chapter is an introduction of the work done. The second chapter is the literature and background review of the work. Chapter three handles methodology, design and implementation of the work done while chapter four is the data presentation and analysis of the work. Chapter five is the conclusion and recommendation of the work for future research and study in this area.

## Chapter Five

### Conclusion And Recommendation

#### 5.1 Conclusion

Accurate loss minimization on high voltage transmission line is a very important factor for the efficiency of electric power network.

In this thesis, mathematical modelling has been applied to study the flow of power on electric power transmission lines. We also utilized the notion of modelling to study power losses on electric power transmission lines.

The classical optimization technique was then used to minimize the power losses on transmission lines thereby proffering solutions to the power loss problem. The application of the classical optimization technique to the mathematical model of power losses on electric power transmission lines provides a better understanding of the problem of power losses on high voltage transmission lines.

#### 5.2 Recommendation

In the course of this research work, we observed that the power loss function is a multivariable function. We treated it as a multivariable optimization problem without constraint.

However, the following are recommended for further investigation and research work.

- There is need to impose constraints on the power loss function to enable precise determination of the optimal value of space between conductors on the transmission lines.
- Empirical models of power losses on transmission lines should be developed as a multivariable function of load, circuit type and distance.
- Models should be formulated for power losses that arises during the cause of generation of electricity. The minimization of this kind of losses should be studied.
- Instead of minimizing power losses along the distribution lines by using feeder reconfiguration and other methods which are capital intensive, similar research should be conducted on distribution systems. Mathematical models should be developed for losses in distribution systems and this kind of models should be minimized using any of the optimization techniques.

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