**TO DEVELOP MATHEMATICAL MODELS FOR POWER LOSSES ALONG TRANSMISSION LINES AND TO MINIMIZE THE LOSSES USING CLASSICAL OPTIMIZATION TECHNIQUES**

**ABSTRACT**

The availability of electric power has been the most powerful vehicle for facilitating any nation’s economic, industrial, and social development. Electric power is delivered to load centers and consumers via transmission lines, which carry bulk power from generating stations to load centers. To ensure that electric power reaches final consumers in proper form and quality, losses along the lines must be kept to a bare minimum. There has been a lot of research done on the analysis and computation of losses on transmission lines using reliability indices, but very little on the minimization of losses using analytical methods. In a similar vein, there is a substantial body of literature for the solution of optimal power flow problems using evolutionary methods, but none of them has used the versatile

A mathematical modeling tool.

Thus, the goal of this work is to minimize transmission power losses by combining the classical optimization approach with the mathematical modeling technique. The study’s specific objectives were as follows:

(i.) create 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 analytically; (ii.) create empirical models of power losses as functions of distance; and (iii.) minimize power losses using the classical optimization technique.

In the study, I used Kirchoff’s circuit laws and a combination xiii of corona and ohmic losses to create mathematical models for power flow and power losses. The empirical models of

Regression analysis was used to develop power losses.

The following were the study’s findings:

The models for power flow along transmission lines evolved as homogeneous second-order partial differential equations that were solved analytically using the Laplace transform method; the model for power losses over transmission lines was obtained as the sum of the ohmic and corona losses; and the empirical models developed are monotonic increasing functions of distance. As a result, power losses increase with distance; and (iv.) power losses are minimized when the operating transmission voltage equals the critical disruptive voltage.

With the above results, a workable strategy for reducing electric power losses along transmission lines to the bare minimum while ensuring electric power availability can be developed.

Consumers are given power in proper form and quality.

As a result, this study addressed the issue of minimizing electric power losses during transmission.

**CHAPTER ONE**

**GENERAL INTRODUCTION**

1.1** BACKGROUND TO THE STUDY**

Energy is a basic requirement for a country’s economic development. There are various types of energy, but the most important is electrical energy. (Gupta, 2008) A modern and civilized society is heavily reliant on electrical energy. Because of the importance of electrical energy to the economic and social development of a society, activities relating to its generation, transmission, and distribution must be given the highest priority in any nation’s national planning process. In fact, the higher a country’s per capita consumption of electrical energy, the higher its people’s standard of living. As a result, a country’s progress is measured in terms of its per capita consumption.

electrical power (Gupta, 2008).

One of the most important and critical issues in power systems is power plant planning to meet power network load demand. Because transmission lines connect generating plants and substations in a power network, scientists and engineers are keenly interested in the analysis, computation, and reduction of transmission losses in these power networks.

Many studies have been conducted on the aforementioned topics.

(Zakariya, 2010) compared the corona power loss and the ohmic power loss associated with HVDC transmission lines. Corona power loss and ohmic power loss were measured and computed for various transmission line configurations and weather conditions. It was mentioned in the work that

The general trend of ignoring corona power loss is not always correct. The comparison revealed that when the transmission line is moderately or lightly loaded, the percentage of corona power loss to ohmic power loss can reach 100%, especially when the transmission line is operating at a voltage well above the corona onset value. This percentage is also found to increase significantly when it rains. Finally, it was discovered that the ratio of corona to ohmic power loss decreases as the number of bundles increases. (Numphetch et al., 2011) investigated loss minimization through optimal power flow using swarm intelligences.

(Thabendra et al., 2009) investigated multi-objective optimization methods for minimizing power loss and maintaining voltage stability. (Abdullah et al., 2010) investigated transmission loss minimization and power installation cost using evolutionary computation for voltage stability improvement. To find the best power system operating conditions, (Bagriyanik et al., 2003) used a fuzzy multi-objective optimization and genetic algorithm-based method. In addition to active power losses, transmission system series reactive power losses were considered as one of the multiple objectives. In his empirical modeling of power losses as a function of line loadings and lengths in the Nigeria 330 KV transmission lines, (Onohaebi, 2010) considered the relationship between distance and loadings on power losses using the existing 330 KV Nigerian transmission network as a case study, while (Moghadam, 2010) developed a new method for calculating transmission power losses based on exact modeling of transmission power losses.

Ohmic loss. (Ramesh et al., 2009) investigated how to reduce power loss in distribution networks through feeder restructuring, distributed generation implementation, and capacitor placement. (Lo, 2006) considered feeder reconfiguration for distribution system loss reduction. Others who investigated power outages include (Rugthaicharoencheep, 2009),

Various researchers, including (Crombie, 2006), have worked on the flow of power on electrical networks. (Pandya, 2008) provides a thorough examination of various optimization methods for solving optimal power flow problems. Linear programming, Newton-Raphson, quadratic programming, nonlinear programming, interior point, and artificial intelligence are among the methods considered in the work. The artificial neural network method, fuzzy logic method, genetic algorithm method, evolutionary programming method, and ant colony optimization were all considered as part of the artificial intelligence method.

method, as well as the particle swarm optimization method. The paper discovered that traditional methods have numerous limitations. Because of the extremely limited capability to solve real-world large-scale power system problems, mathematical formulations must be simplified in most cases to obtain solutions. The classical methods are ineffective at dealing with qualitative constraints and have poor convergence. In addition, when dealing with large-scale optimal power flow problems, the methods are very slow and computationally expensive. The paper also discovered that artificial intelligence methods are relatively versatile for dealing with various qualitative constraints and can find multiple optimal solutions in a single simulation.

As a result, they are appropriate for solving multi-objective optimization problems.

(William, 2002) investigated alternative optimal

While (Claudio et al., 2001) worked on comparing voltage security constraints using optimal power flow techniques.

Roya et al. (2008) investigated power flow modeling in power systems with dynamic flow controllers. Others who have worked on power flow include: (Bouktir et al., 2004).

Furthermore, several researchers have worked on electric power systems. (Aderinto, 2011) developed an optimal control model for a power generation system. She developed a mathematical model for the electric power generating system using the optimal control approach in her research work, and she characterized the mathematical model by prescribing the conditions for the optimality of the electric power generating system as well as the analytic requirements for the existence and uniqueness of the system’s solution. The model’s optimality condition was determined, and the model was solved analytically and numerically. Two control variables were identified in the study, the first for load shedding among the generators in the system and the second for generator capacity restriction. Because the first control variable can only be on or off, the problem was formulated using the second control variable. The system’s optimality conditions were implicitly imposed on the controls, and the mathematical model represents a stable loss-free generating system. The work demonstrated that generation loss can be controlled and stabilized. (Oke et al., 2007) examined Nigerian perspectives on electricity supply and demand, whereas (Ibe,

2007) investigated optimal electricity generation in Nigeria. (Bamigbola, 2009) described an optimal control model for a power generation system. (Karamitsos, 2006) investigated blackout analysis for electric power transmission systems, whereas (Aderinto et al., 2010) investigated optimal control of air pollution with application to a power generation system model. Others who have conducted research on electric power systems include (Savenkov, 2008), to name a few. As a result, much emphasis has been placed on proper electrical power system design and loss reduction through feeder reconfiguration and evolutionary techniques.

Loss reduction is an important component of efficient electric power supply systems.

Electric power system losses should be in the range of 3% to 6%. (Ramesh et al., 2009). It is not more than 10% in developed countries. However, it is still higher than 20% in developing countries (Ramesh et al., 2009). As a result, power sector stakeholders are currently interested in reducing losses on electric power lines to a desired and economical level. The goal of this research is to create mathematical models for power losses along transmission lines and then use classical optimization techniques to minimize the losses.

**1.2 OBJECTIVES OF THE STUDY**

Power losses reduce the amount of power available to consumers, resulting in insufficient power to operate their appliances. Low power losses characterize a high-efficiency power system. The goal of this research is to minimize transmission power losses on transmission lines using classical optimization techniques. The research work’s objectives are as follows:

(i.) Create mathematical models for electric power flow and power losses along transmission lines;

(ii.) Analyze the mathematical models for electric power flow along transmission lines.

(iii.) Create empirical models of power losses as they relate to distance.

(iv.) Using the classical optimization technique, minimize power losses.

**1.3 SIGNIFICANCE OF THE STUDY**

The mathematical representation of power flow along transmission lines improves understanding of power flow on transmission lines as well as the evolution of voltage and current along the lines. The mathematical representation of power losses along transmission lines provides insight into the major transmission problems.

The use of classical optimization techniques to minimize losses on electric power transmission lines provides a compact solution to the major problem encountered in power transmission.

**1.4 ORGANIZATION OF THE THESIS**

There are five chapters in this dissertation report. The first chapter is an overview of the work completed. The second chapter is a review of the work’s literature and background. Chapter three deals with methodology, design, and implementation of the work, while Chapter four deals with data presentation and analysis. The fifth chapter is the work’s conclusion and recommendations for future research and study in this area.

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