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1.1     Background of the Study

For many years, the demand for and consumption of energy in many countries of the world has been on the increase. The major portion of the energy needs of these nations is electric energy. In Nigeria and other industrial developing nations, the demand for supply of electrical power has been on the increase, which may be as a result of improved economic activities of the people. To satisfy the increasing demand for electricity, complex power system networks have been built. The most usual practice in electric power transmission and distribution is an interconnected network of transmission lines usually referred to as a grid system that links generators and loads to form a large integrated system that spans the entire country. In many countries of the world including Nigeria, generating stations are located thousands of kilometers apart and operate in parallel. The generating stations’ output is connected and transmitted through the grid system to load centers nationwide.

The complexity of an interconnected electric power system network provides different challenging engineering problems to the operators. These problems are in aspect of planning, construction, operation and control of the system. The problems can stimulate the managerial talent of the operator, while others tax his knowledge and experience in network design.

One operating characteristic of power systems is that the devices included in the model can reach a particular state at which the equation that models the system will change. This characteristic is independent of many of the other assumptions used to model the system. The new state of the network must be predicted on automatic control and not on human operational response that can be very slow. The operator is forced to rely on ever more powerful tools of solving the problem of prediction of the performance of the complex system. One of the several problems confronting the efficient performance of an interconnected system is voltage stability.







Voltage stability issues are of major concern worldwide because of the significant number of black-outs that have occurred in recent times in which it was involved. For many power systems, assessment of voltage stability and prediction of voltage instability or collapse have become the most important types of analysis performed as part of system planning, operational planning and real-time operation. Voltage stability is defined as the ability of a power system to maintain steady acceptable voltages at all buses in the system under normal operating conditions, and after being subjected to a disturbance [1]. In other words, voltage stability is the ability of a system to maintain voltage so that when load admittance is increased, load power will increase, and so that both power and voltage are controllable. The ability to transfer reactive power from production sources to consumption sinks during steady operating conditions is a major aspect of voltage stability. Voltage stability deals with the ability to control the voltage level within a narrow band around normal operating voltage.

The consumers of electric energy are used to rather small variations in the voltage level and the system behaviour from the operators’ point of view is fairly well known in this normal operating state. Equipment control and operation are tuned towards specified set points giving small losses and avoiding power variation due to voltage sensitive loads.

Once outside the normal operating voltage band many things may happen of which some are not well understood or properly taken into account today. A combination of actions and interactions in the power system can start a process which may cause a completely loss of voltage control. It is known that to maintain an acceptable system voltage profile, a sufficient reactive support at appropriate locations must be found. Nevertheless, maintaining a good voltage profile does not automatically guarantee voltage stability. On the other hand, low voltage although frequently associated with voltage instability is not necessarily its cause [2] and [3].

Voltage stability studies of a power system is now essential and is intended to help in the classification and the understanding of different aspects of power system stability [4].

Voltage stability evaluation requires the determination of:-

  • The parameters and a stress test that establish the structural causes of voltage collapse and instability in each load area (exhaustion of reactive reserves in a reactive reserve basin).


  • A method of identifying each load area (voltage collapse and instability area) that has a unique voltage collapse and instability problem, and
  • A measure of proximity to voltage collapse for each load area (a measure of reactive reserve or voltage control areas with zero reserves in the reactive reserve basin)

Voltage stability assessment involves two methods, the static method and dynamic method. The static method is intended to evaluate the voltage stability margin based on load flow steady state analytical techniques, such as continuation method, multiple power flow solutions, sensitivity analysis, singular value  or modal analysis of Jacobian matrix, etc. It consists of either load flow or steady state stability methods. Static analysis is useful for indicating the possibility of voltage collapse.

The dynamic method employs non-linear algebraic and differential equations in the power system model. It indicates the true dynamic behaviour of the voltage instability. Dynamic analysis is important in complementing the steady state analysis and for a better understanding of voltage stability phenomena.

One of the operating goals of an electric power system is to attend the power demand keeping the system’s voltages as well as the frequency close to rated values. Deviation from these nominal conditions may result in abnormal performance of or even damage to the supplied equipment. An unacceptable voltage level means voltage instability. The voltage instability, also known as voltage collapse of power systems appears when the attempt of load dynamics to restore power consumption is just beyond the capability of the combined transmission and generator system [5]. The problem is also a main concern in power system operation and planning. It is characterized by a sudden reduction of voltage on a set of buses of the system. In the initial stage the decrease of the system voltage starts gradually and then decreases rapidly.

The following can be considered the main contributing factors to the problem [6]:

  1. Stressed power system; i.e., high active power loading in the system.
  2. Inadequate reactive power resources.
  3. Load characteristics at low voltage magnitude and their difference from those traditionally used in stability studies.



  1. Transformers tap changer response to decreasing voltage magnitudes at the load buses.
  2. Unexpected and or unwanted relay operation may occur during conditions with decreased voltage magnitudes.

This problem is a dynamic phenomenon and transient stability simulation may be used. However, such simulations do not readily provide sensitivity information or the degree of stability. They are also time consuming in terms of computers and engineering effort required for analysis of results.

The problem regularly requires inspection of a wide range of system conditions and a large number of contingencies. For such application, the steady state analysis approach is much more suitable and can provide much insight into the voltage and reactive power loads problem [7] and [8].

So, there is a requirement to have an analytical method, which can predict the voltage collapse problem in a power system. As a result, considerable attention has been given to this problem by many power system researchers. A number of techniques have been proposed in the literature for the analysis of this problem [9].

The problem of reactive power and voltage control is well known and is considered by many researchers.

The dynamic analysis is especially critical in the final stages at the points of voltage collapse.

Dynamic voltage stability is analyzed by monitoring the eigen-value of the linearized system as a power system is progressively loaded. Instability occurs when a pair of complex eigen-value crosses to the right half plane. This is referred to as dynamic voltage instability.

The system will experience a voltage collapse and this will result to a rapid loss of electrical supply in wide areas, sometimes affecting millions of people.

The origin of a significant voltage deviation is in most cases some kind of contingency where generation in a vital power plant shuts down or an important transmission line is disconnected from the power grid.

This indicates a voltage change and alters the system characteristics. The system is normally designed to withstand these kinds of single contingencies occurring many times


a year. However, abnormal operating conditions, several independent contingencies occurring almost simultaneously in time or a completely unexpected phenomenon may violate the normal design conditions. These lead to an insecure operating condition threatening the voltage stability of the system. The goal is therefore to try to understand the course of events after such a contingency and propose remedial actions when the control of voltage is insecure.

In this thesis, a powerful software NEPLAN operated in matlab 7.5.0 environment is used to model an interconnected power system networks using power system analysis tools (PSAT) and simulated for voltage stability evaluation using Modal Analysis Technique.

This technique was chosen after reviewing all available literature presented in Chapter two of this thesis, and found that (Modal analysis Technique) is capable of determining the objectives (i) – (iii) mentioned above with less stress to the researcher [10]

The Modal analysis calculates the smallest eigenvalues of the reduced Jacobian Matrix  and the bus, branch, and generator participation factors. The smallest eigenvalue and its associated eigenvectors of   at the nose of the PV curve defined the critical mode of voltage stability. The corresponding bus, branch and generator participations identify the voltage stability sub-zone, and the elements that have large impact on the voltage stability of this critical mode. This would enable remedial measure to be applied at the sub-zone identified by these participations so as to enhance the voltage stability of the critical zone and mitigate the negative impact of these elements on the overall system voltage stability.

1.2     Statement of the Problem

The analysis of voltage stability for a given system involves the examination of two aspects:

(1) Proximity to voltage instability.

  • How close is the system to voltage instability (which is the distance to instability)

This may be measured in terms of physical quantities, such as load level, active power flow through a critical interface, and reactive power reserve. The most appropriate measure for any given situation depends on the specific system and the



intended use of the margin; for example, planning versus operating decisions. Also in measurement, consideration must be given to possible contingencies (line outages, loss of a generating unit or a reactive power source, etc.).

(11) Mechanism of voltage instability.

In considering the mechanism of voltage instability the following issues must be clarified:

  • How and why does instability occur?
  • What are key factors contributing to instability?
  • What are the voltage weak areas?
  • What measures must be effected in improving voltage stability?

Time-domain simulation, in which appropriate modelling is included, capture the events and their chronology leading to instability. However, such simulations are time-consuming and do not readily provide sensitivity information and degree of stability. System dynamics influencing voltage stability are usually slow. The static analysis like Modal/Eigenvalue technique can provide much insight into the nature of the problem and identify the key contributing factors. The advantage of Modal analysis technique is that it gives voltage stability-related information from system –wide perspective and clearly identifies areas that have potential problem. It has the added advantage that it exposes and measures all masked information regarding the mechanism of instability.

The analysis of a generator connected to an infinite bus does not pose the kind of challenge encountered when an interconnected system is in focus. The challenge is always associated with modelling capability of the interconnected system and the ability to solve the problem using simulation software operated in the Matlab environment as proposed in this thesis.

1.3     The Research Motivation

An interconnected power system is affected by events that depend upon the states

(voltage magnitude and current) and parameters ( real and reactive power) of the electric power system. The states and parameters of the power system are influenced by both controllable and uncontrollable factors. The problem of voltage stability has been addressed by numerous papers published around the world that discussed ways to tackle this pressing problem. For some years, the increasing higher power demands and the


restricted growth of electric transmission systems have forced utilities to operate power networks close to their transmission limit, this has created new voltage stability problems. While some forms of disturbances resulting in changes in reactive power demand may trigger the process of voltage instability, the causes of stressed systems are many [6]. The high cost of upgrading and strengthening existing transmission lines to meet the increasing demand for electricity, inadequate provision of new generating plants, the difficulty of acquiring way-leaves from the rural dwellers are just a few reasons that lead to the increase of vulnerability in today’s interconnected power system network. Power system operators are researching to enhance their understanding of where the system is operating with respect to the point of collapse. This point is often referred to as the critical point. The identification of the critical point indirectly defines the boundary between the stable and unstable steady state operating region, the research for the critical point is of importance. However, non-linearity gives rise to complex and unexpected behaviour for many physical, chemical and biological systems. An interconnected power system is not an exception for this complicated behaviour. A particular feature seen in many nonlinear systems is the abrupt change in steady-state behaviour that may occur as a parameter changes smoothly. The parameter values for which the abrupt change occurs generally correspond to singular points in the governing equations. At a singular point, the Jacobian matrix of the linearized system is singular and one real eigenvalue for the linearized system crosses the imaginary axis at such a point. This is accompanied by qualitative change in behaviour. In power systems, it can be related to voltage instability/collapse.

1.4 Purpose of the Study

From the points raised in section 1.3, it is clear that the state of the power system is influenced by both controllable and uncontrollable factors. For example, the system generation must increase as load increases to keep power balance and maintain voltage stability. Often at high load levels, generators reach real and reactive power limitation and the power flows in lines exceed limits. Any of these events can initiate a change in the equations that model the power equation. This thesis is intended to achieve the following aims and objectives:




  • To model an interconnected electric power system network that will continually track system changes and assess its voltage stability.
  • To predict the possible causes of voltage collapse in an interconnected electric power systems like the Nigerian power system.
  • To access stability margin and power transfer limit in the system
  • To indicate sensitivities and the major contributing factors that will provide insight into system characteristics to assist in developing remedial actions.

1.5     Scope of Study

Many researchers in voltage stability have proposed and adopted various techniques in solving the problem of an interconnected electric power system network. This research is limited to Modal/Eigenvalue analysis technique, which has been successfully applied to many international interconnected power system networks and was shown to be very efficient [7]. However, the Q-V Sensitivity, Q-V Curves, and participating factors were used to confirm the results obtained from the analysis and to predict the stability margin based on reactive power demand.

The modal/eigenvalue analysis technique, Q-Vsensitivity, Q-VCurves, were implemented using NEPLAN Simulation software implemented using power system analysis tools (PSAT) and Matlab program. In this thesis, plots were extensively used to present the results. The relevance is that with plots it is easier to identify patterns than tables of numbers. Researchers usually use plots both to gain insight and present their research findings and ideas to others. Two systems that were modelled and simulated in this thesis include an international sample test network, the IEEE 14 Bus system and a Nigerian interconnected 330kV , 30Bus electric power system [11]. These systems were shown in Figure 1.1 and Figure 1.2 respectively. The Nigerian System was also compensated and studied.




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