A Power System Stabilizer Engineering Essay

Published: Last Edited:

This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.

This report describes damping of power system oscillations using two state feedback controllers developed by Electricité de France. Damping of power system oscillations is essential as these oscillations are highly undesirable. The controllers are named as the DFLR (Desensitized Four Loop Regulator) and EDFLR (Extended Desensitized Four Loop Regulator). The DFLR is designed to damp local electromechanical oscillations while the EDFLR aims at damping both local and inter-area oscillations.

Power systems contain many modes of electromechanical oscillation as a result of interactions of its components, for example one generator rotor swings relative to one another. These electromechanical oscillations cause oscillations in state variables of the power system (voltage, current, power and frequency of the nodes of the system).

There are two major types of oscillations which have proved troublesome in power systems:

Local oscillations occur when a generator or a group of generators under voltage regulator control at a station is swinging against the rest of the system. The oscillations of this kind are in the range of 1Hz to 2 Hz.

Inter-area oscillations involve combinations of many synchronous machines on one part of a system swinging against synchronous machines on another part of the system.

The characteristic frequency of inter-area modes of oscillation is usually in the range of 0.1 to 0.6 Hz. These are much more difficult to damp compared to the local modes of oscillations.

The classical, most cost effective and widely used process to damp power system oscillations is the installation of Power System Stabilizer (PSS) alongwith the Automatic Voltage Regulator (AVR).


Automatic Voltage Regulator is a device which controls and stabilizes the terminal voltage output of the generator. It is designed in such a manner to achieve good time response when the generator is disconnected from the network. Static AVR has a destabilizing effect on electromechanical oscillations. Whenever the terminal voltage tends to fall due to load, the drop is immediately detected & excitation is increased thereby the required terminal voltage is maintained in the output terminals.


A POWER SYSTEM STABILIZER (PSS) is installed in the Automatic Voltage Regulator of the Generator. It is mainly designed to improve the power system stability. Therefore the PSS has excellent cost performance. It can be observed as an additional block of a generator excitation control, which is added to improve the overall power system dynamic performance, especially for the control of electromechanical oscillations. Auxiliary stabilizing signals such as shaft speed, terminal frequency or power is used by the PSS to change the input signal to the AVR. This is a very effective method of enhancing small-signal stability performance of a power system network. The generator output is decided by the turbine mechanical torque. But its output power also can be changed by changing excitation value transiently. The changing of generator output power is detected and the excitation value is controlled by the PSS. The use of Power System Stabilizers has become increasingly important as it provides improved stabilization of the system.

Figure showing how a tuned PSS aims at faster damping of oscillations


Local Mode Power Oscillation

Individual generator oscillates against the system.

Frequency is approx. 1 Hz.

PSS of single input

Inter-area (Long Cycle) Mode Power Oscillation

The whole system oscillates with long distance and large power transportation system connection.

Frequency is 0.2 to 0.5 Hz

PSS of single input


The objective of this project is to implement the DFLR and the EDFLR into the power system model to demonstrate the use of these controllers at damping the electromechanical oscillations occurring in the power system. As we had already seen that the power system electromechanical oscillation phenomenon is highly undesirable for effective operation of a power system, it is required to damp these oscillations efficiently and effectively. Before applying these controllers in any power system it is beneficial to simulate the power system along with the controllers in order to get a beforehand knowledge about how effectively these controllers can work when properly tuned. For that purpose we need to mathematically model these controllers by studying their dynamics. Then we develop the generator models and by combining these we get the complete closed loop power system model with state feedback control.

Chapter 2








In state feedback control, the state variables are fed back to the input of the system instead of the output variables. State variables are those system variables that summarizes the past that is useful for prediction.

This technique can only work if the system is controllable. The future behavior can be predicted from the state if all the states are known. This mode of feedback control is easy to implement and has a wider range of applications.

Fig 2.1 State feedback control model


The AVR and the PSS were traditionally designed from the transfer function representation of the system and its frequency response. The DFLR (Desensitized Four loop Regulator) and the EDFLR (Extended Desensitized Four loop Regulator) are the stable state feedback controllers aimed at damping power system oscillations. The DFLR is aimed at damping local oscillations. It is a robust controller and it offers good performance in spite of variations in generator operating conditions. It comes from the state space form of single machine infinite bus system. The three state variables are terminal voltage, active power and speed deviation. It feeds back the three states multiplied by the corresponding gains. The gains are obtained by the desensitization method based on optimal control theory. The EDFLR is aimed at damping both local and inter-area oscillations. It is the result of the application of an extended version of the desensitization method to a more complicated design circuit with two generators.



Fig 2.2 Single machine infinite bus system - Design circuit of DFLR

The state feedback structure arises from the state space representation of the design circuit as shown in figure 2.2. The circuit represents the local oscillatory phenomenon through the single electromechanical oscillation of the generator against the infinite bus. The figure 2.4 presents a scheme of the state-space model. The terminal voltage V, the connection reactance X, the active power P and the reactive power q describe the operating conditions. The control input is the excitation voltage Efd . The measured outputs are the terminal voltage V, the active power Pe and the speed deviation ω. To eliminate the steady state error between the terminal voltage and reference voltage, integral action is included in the voltage regulation loop.

Ä- =Vref - Vt

The resulting state feedback controller with integral gain is obtained whose general structure is shown the figure below.

Fig 2.3 State space model of DFLR

Fig 2.4 Four loop regulator structure

This controller obtains the signal Efd as the sum of each of the three measured variables multiplied by a gain.

The four gains are calculated according to the desensitization method. As this controller provides simultaneously both the voltage regulation given by AVR and damping to local oscillations corresponding to PSS, it is equivalent to a coordinate AVR + PSS tuning.


Fig 2.5 Design circuit of EDFLR

The design circuit is a two generator circuit aimed at representing two electromechanical oscillations:

1] a local mode of G1 against G2 mainly controlled by Xlo

2] an inter-area mode of G1 and G2 against the infinite bus controlled mainly by Xin.

The apparent power S2 controls the participation of G1 in the inter-area mode. The frequencies of the local and inter-area modes and the participation factor of the generator of the inter-area mode are the parameters of the design circuit. These parameters are retrieved from the detailed model of the real system where the controller is to be installed.


Fig 2.6 Standard AVR + PSS structure

The input signal used for the PSS is the accelerating power Pa = Pm - Pe

The control signal E fd obtained from the standard structure where Vref is zero since it is assumed that no change in the reference voltage occurs. All variables and functions are expressed by their Laplace transform.

……………………………… (1)

If E fd (s) is obtained from the four loop regulator structure then

………………….. (2)

The four gains are represented by transfer functions to consider the general case associated with the EDFLR. The main purpose is to obtain the transfer functions AVR(s) and PSS(s) as a function of these four gains. Hence, all measured variables should be expressed as a function of V t and Pa .

The integration error can be expressed in terms of the terminal voltage

The speed deviation is related to the accelerating power by the swing equation of the rotor

As neither the AVR nor the PSS transfer functions have a pure integrator, the integrators of error and speed deviation are approximated by the first order transfer function. At low frequencies, the infinite gain of the integrator is limited to T but if T is sufficiently high, the crossover frequency is low enough to be out of the frequency range of interest (0.1 Hz - 10 Hz).

Approximating the integrators of error and speed deviation by the first order transfer function with time constants of respectively T AVR and T PSS and substituting the resulting expressions in (2), Efd (s) is given in terms of V t (s) and Pa (s)

…… (3)

From the comparison of (1) and (3), the transfer functions of the AVR and PSS are

Thus we have obtained the gains AVR (s) and PSS (s) in terms of the four gains. Hence by the knowledge of these four gains we can simplify the controllers to the standard form.


The gains are obtained as Kvt =31.202, Kpe= 13.417, Kw=-2.387, Ke=12.788 by desensitization method [1], The following figures 2.7 and 2.8 can be obtained using these gains [1]


The AVR is composed of a gain of 200, the delay of the static exciter (a time constant of 0.0033 s) and a transient gain reduction block (a lag network) to reduce the steady state error of the voltage. The PSS is formed by a washout filter (with a time constant of 5 s), a lag phase compensation network, and the PSS gain .


Chapter 3







The basic techniques for studying the stability of interconnections of synchronous generators stem from the late nineteenth century and the early years of last century. The key concept of transforming stator variables into quantities rotating in synchronism with the rotor was developed by Blondel, Park, and others and remains the basis for synchronous machine analysis to this day.

To some extent, the techniques developed in those early years remained relatively untouched until the last three or four decades of the twentieth century. Although it was in theory possible to develop relatively complex generator models prior to this time, limited computational capability meant that such models were impractical for use in large-scale stability studies. However, with the advent of the digital computer, the picture changed significantly and computational capability continues to grow at a rapid rate. In addition, the growing complexity of electric power systems combined with the advent of more sophisticated generator and system controls greatly increased the demands on stability programs.



Traditionally, power system stability studies focuses on the system's ability to maintain synchronous operation following a severe disturbance. However, with continuing growth in interconnections, more use of new technologies, and the increased need to operate power systems in highly stressed conditions, other forms of stability have emerged. Clearly, instability in a power system may be evidenced in many different ways depending on the system configuration, operating mode, and form of disturbance.

Analysis of stability problems, including identifying essential factors that contribute to instability and devising methods of improving stable operation, is greatly facilitated by classification into appropriate categories. These are based on the following considerations:

- The physical nature of the resulting instability

- The size of disturbance considered, impacting on the applicable method of analysis.

Based on the physical nature of the phenomena, power system stability may be classified into three main categories:

(a) Rotor-angle stability

(b) Voltage stability

(c) Frequency stability


Rotor-angle stability is concerned with the ability of interconnected synchronous machines of a power system to remain in synchronism under normal operating conditions and after being subjected to a disturbance. A fundamental factor in this aspect of stability is the manner in which the torque or power outputs of the synchronous machines vary as their rotors oscillate. The mechanism by which synchronous machines maintain synchronism with one another is through the development of restoring torques whenever there are forces tending to accelerate or decelerate the machines with respect to each other.

The change in electromagnetic torque of a synchronous machine following a perturbation can be resolved into two components:

a synchronizing torque component, in phase with the rotor-angle deviation

a damping torque component, in phase with the speed deviation.

The lack of sufficient synchronizing torque results in aperiodic instability, whereas the lack of damping torque results in oscillatory instability. Loss of synchronism can occur between one machine and the rest of the system, or between groups of machines, possibly with synchronism maintained within each group after separating from each other.

For convenience in analysis and for gaining insight into the nature of stability problems, it is useful to characterize rotor-angle stability into the following subcategories based on the size of disturbance considered:

a) Large-disturbance angle stability, commonly referred to as transient stability, is concerned with the ability of the power system to maintain synchronism when subjected to a severe disturbance, such as a transient fault on a transmission circuit, or loss of a large generator. The resulting system response involves large excursions of generator rotor angles and is influenced by the nonlinear power-angle relationship of synchronous machines. Usually, the disturbance alters the system such that the post-disturbance conditions will be different from those prior to the disturbance. Instability is in the form of an aperiodic drift of the rotor angle due to insufficient synchronizing torque. In large power systems, transient instability may not always occur as first swing instability associated with a single mode; it could be the result of increased peak deviation caused by superposition of several modes of oscillation causing large excursions of rotor angle beyond the first swing.

b) Small-disturbance angle stability is concerned with the ability of the power system to maintain synchronism under small disturbances such as those that continually occur in the normal operation of the power system. The disturbances are considered to be sufficiently small that linearization of system equations is permissible for purposes of analysis. Small-signal analysis using linear techniques provides valuable information about the inherent dynamic characteristics of the power system. Instability that may result can be of two forms:

(i) increase in rotor angle through a non-oscillatory or aperiodic mode due to lack of synchronizing torque

(ii) rotor oscillations of increasing amplitude due to lack of sufficient damping torque.

In present-day power systems, the small-disturbance angle stability problem is usually one of insufficient damping of oscillations. The stability of the following types of oscillations is of concern:

- Local-plant-mode oscillations, associated with units in a power plant swinging against the rest of the power system.

- Inter-area-mode oscillations, associated with the swinging of a group of generators in one area against a group of generators in another area.

- Torsional-mode oscillations, associated with the turbine-generator shaft system rotational components of individual generators.


Voltage stability is concerned with 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. Instability that may result occurs in the form of a progressive fall or rise of voltage of some buses with only moderate deviation of generator angles. The main factor causing voltage instability is the inability of the power system to maintain a proper balance of reactive power throughout the system. This is significantly influenced by the characteristics of system loads and voltage control devices, including generators and their excitation system.

It is useful to classify voltage stability into the following two subcategories based on the size of disturbance considered:

a) Large-disturbance voltage stability is concerned with a system's ability to maintain steady voltages following severe disturbances. The evaluation of stability usually requires the examination of the dynamic performance of the power system over a period of time sufficient to capture the interactions of such devices as under-load transformer tap changers and generator field-current limiters. The study period may extend from a few seconds to several minutes.

b) Small-disturbance voltage stability is concerned with a system's ability to maintain steady voltages following small perturbations, such as incremental changes in load.


Frequency stability is concerned with the ability of a power system to maintain the frequency within a nominal range. It depends on the ability to restore balance between system generation and load with minimum loss of load. Analysis of frequency stability is carried out using time-domain simulations that include all appropriate fast and slow dynamics sufficient for modeling the control and protective systems that respond to large frequency excursions as well as the accompanying large shifts in voltages and other system variables.

In the case of large interconnected power systems, simulations required may include severe disturbances beyond the normal design criteria, which result in cascading and splitting of the power system into a number of separate islands with generators in each island remaining in synchronism. Stability in this case is a question of whether or not each island will reach an acceptable state of equilibrium with minimum loss of load.


Synchronous machines may be modeled in as much detail as possible in the study of most categories of power system stability. This includes appropriate representation of the dynamics of the field circuit, excitation system, and rotor damper circuits. For the analysis of many voltage-stability and frequency-stability problems using time-domain simulations, the study periods are in the range of tens of seconds to several minutes. To improve computational efficiency of such long-term dynamic simulations, instead of simplifying the models by neglecting fast dynamics, it is better to use singular perturbation techniques to separate fast and slow dynamics and appropriately approximate the fast dynamics.

The following special requirements in representing synchronous machines for different categories of stability studies are:

a) For large-disturbance rotor-angle stability analysis, particularly for generators with high-initial response excitation systems, magnetic saturation effects should be accurately represented at flux levels corresponding to normal operation all the way up to the highest values experienced with the excitation at its peak level. With discontinuous excitation controls, the excitation remains at its peak for about two seconds leading to very high flux levels. If saturation effects are downplayed, the results of analysis would be overly optimistic. It is important to represent the dynamics of the field circuit, as it has a significant influence on the effectiveness of excitation system in enhancing large-disturbance rotor-angle stability.

b) For small-disturbance rotor-angle stability analysis, accurate representation of the field circuit as well as the rotor damper circuits is important.

c) For voltage stability studies, the voltage control and reactive power supply capabilities of generators are of prime importance. During conditions of low system voltages, the reactive power demand on generators may exceed their field-current limits. In such situations, usually the generator field currents are automatically limited by over-excitation limiters, further aggravating the situation and possibly leading to voltage instability. Therefore, the generator models should be capable of accurately determining the transient field currents and accounting for the actions of field current limiters.

d) Frequency stability problems are generally associated with inadequacies in equipment response and poor coordination of control and protection equipment. Stability is determined by the overall response of the system as evidenced by its mean frequency, rather than relative motions of machines. The generator models used should be capable of accurately representing, under conditions of large variations in voltage and frequency, the responses of control and protective devices, such as the voltage regulator, power system stabilizer, V/Hz limiter and protection, and over-excitation and under-excitation limiters.


Synchronous generators are most commonly constructed with a three-phase armature winding on the stator and an excitation winding (known as the field winding) on the rotor. In addition, synchronous generator rotors include other conducting paths in which currents can be induced during a transient. In some cases, these conducting paths are intentionally included by the designer like the pole-face damper windings. In other cases, they are inherent to the machine design, like the currents which can be induced in the rotor body of a solid-rotor turbo generator.

Earlier in the process of developing techniques for the analysis of synchronous machines, it was recognized that analyses can be greatly simplified if they are performed in a reference frame rotating with the rotor. For such analyses, the armature currents and voltages are transformed into two sets of orthogonal variables, one set aligned with the magnetic axis of the field winding, known as the rotor direct axis (d-axis), and a second set aligned along the rotor at a position 90 electrical degrees from the field-winding magnetic axis. This second axis is known as the rotor quadrature axis (q-axis).

Much of the simplification associated with such an approach stems from two key features:

1) Under steady-state operating conditions, all of the currents and fluxes, including both those of rotor windings and the transformed armature windings, have constant values.

2) By choosing the two axes 90 electrical degrees apart, fluxes produced by currents in the windings on one axis do not produce flux linkages in the windings on the other axis. Thus, these sets of windings are orthogonal.

This greatly simplifies the flux-current relationship of the model and gives rise to a model structure consisting of two independent networks, one for the direct axis and one for the quadrature axis.


The direct axis of a synchronous machine includes two terminal ports. These correspond to the direct-axis equivalent armature winding and the field winding. An accurate representation of the direct axis must fully account for the characteristics of both of these terminals. The simplest direct-axis representation assumes that there are no other current paths in the direct axis other than the direct-axis armature winding and the field winding. However, it is well known that damper-winding currents (in the case of salient-pole machines) or rotor-body currents (in the case of solid-rotor machines) play a significant role in determining the characteristics of the direct axis. Hence, the most common direct axis model includes an additional winding, known as the direct-axis damper winding.

Part (a) of Figure 3.1 shows the equivalent-circuit representation for the direct-axis model with a single damper winding. This equivalent circuit includes an ideal transformer, representing the fact that there are differing numbers of turns on the armature and field winding, just as is the case for the primary and secondary windings of a transformer.

using ideal transformer

without ideal transformer

Fig 3.1 D-axis equivalent circuits

It is common to represent synchronous machines using a per-unit representation, rather than actual units, in which case an ideal transformer may or not be required, depending upon the choice of the base for the per unit system. Whether in actual units or in per unit, the ideal transformer is typically left out of the equivalent circuit, resulting in the equivalent circuit of part (b) of Figure 3.1 in which the field voltage and current are as reflected to the armature winding.


Because there is no rotor winding with terminals on the quadrature axis, the quadrature axis need be represented only as a single-port network. In addition to the quadrature-axis armature winding, varying numbers of damper windings can be included in the quadrature-axis model. The flux-current relations for the quadrature-axis models are directly analogous to those presented earlier for the direct-axis.

Fig 3.2 Q-axis equivalent circuit


The simplest model that can be used to represent a synchronous machine is by a constant voltage and a single series reactance representation. In the steady state, this representation includes the synchronous reactance and the voltage behind synchronous reactance, which is proportional to the field current supplied to the generator. In this representation, saliency is neglected and the synchronous reactance is set equal to the direct-axis synchronous reactance of the machine. The transient model of this type is assumed to be valid for the initial time period of an electromechanical transient and can be used to roughly estimate the first-swing stability of a synchronous machine.

During a transient simulation, the magnitude of the model's internal voltage is kept constant, but the internal angle is changed corresponding to the rotational dynamics of the generator rotor. Advantages of this simple model are that the interfacing of the generator and network equations can be accomplished more quickly during transient simulations and that it requires relatively little data.



Transient stability analysis involves the analysis of power system performance when subjected to a severe fault. Power systems are designed and operated so as to be stable for a set of contingencies referred to as the design contingencies. These contingencies are selected on the basis that they have significant probability of occurrence given the large number of elements comprising the power system.

In transient-stability studies, the important issues are:

- Calculation of generator power or torque during the fault period.

- Calculation of post-fault generator power, angle, and voltage for a period of up to several seconds after the fault cleared.


Small-disturbance stability analysis involves the examination of the ability of the power system to maintain synchronous operation when subjected to small perturbations. The modeling of the power system, including synchronous generators, is very much similar to that for transient stability analysis. Only balanced operation is considered and the system equations may be linearized for purpose of analysis. Generator models used for small-disturbance stability assessment should accurately account for damper circuit effects, field circuit dynamics, and excitation control. One of the effective means of enhancing small-signal stability is the use of power system stabilizers



Salient-pole generators with laminated rotors are usually constructed with copper-alloy damper bars located in the pole faces. These damper bars are often connected with continuous end-rings and thus, form a squirrel-cage damper circuit that is effective in both the direct axis and the quadrature axis. The damper circuit in each axis may be represented by one circuit. Salient-pole machines with solid-iron poles may justify a more detailed model structure with two damper circuits in direct axis.


In cylindrical-rotor machines, slots are present over part of the circumference to accommodate the field winding. The tops of these slots contain wedges for mechanical retention of the field turns. These wedges are usually made of a nonmagnetic metal, and may be either segmented or full length. In many constructions, a conductive ring under the field end-winding retaining ring, with fingers extending under the ends of the slot wedges, is used to improve conduction at these connection points. Copper strips are often inserted under the wedges to provide improved conduction between wedge segments and to improve damper-circuit action. In some cases, a complete squirrel-cage winding is formed, while in other cases the conductive paths contribute only marginally to damper-circuit action.



A simplification to the synchronous generator model is to neglect the effects of damper circuits. The primary reason for this approximation is that often machine parameters related to the damper circuits are not readily available, particularly for older units. Neglect of damper circuits' effects introduces some degree of loss of accuracy. Depending on the nature and scope of the study, this may be acceptable.


The classical model offers considerable computational simplicity as it allows the transient electrical performance of the generator to be represented by a simple voltage source of fixed magnitude behind an effective reactance. Such a model is now used for screening studies, such as contingency screening and ranking for transient stability limit search applications. The classical model is also used for representing "remote" machines in the analysis of very large interconnected power systems.



Mathematical models of a synchronous machine vary from elementary classical models to more detailed ones. The detailed models are transient and subtransient models. The following equations link the mechanical variables with the electrical variables, and result in the block diagram

representation depicted in Figure 3.3:


D = the damping constant

Ï„j = the inertia time constant

Tm = the input mechanical torque

E′′d = subtransient generated voltage in the direct axis

E′′q = subtransient generated voltage in the quadrature axis

Id = the armature current in the direct axis

Iq = the armature current in the quadrature axis

For eigenvalue or time domain simulation studies, it is necessary to include the effects of the excitation controller, which indirectly controls the reactive output of a generator

Machine subtransient model

Chapter 4





Simulink is a software package for modeling, simulating, and analyzing dynamic systems. It supports linear and nonlinear systems, modeled in continuous time, sampled time, or a hybrid of the two. Systems can also be multirate, i.e., have different parts that are sampled or updated at different rates. For modeling, Simulink provides a graphical user interface (GUI) for building models as block diagrams, using click-and-drag mouse operations

With the help of MATLAB 2009a and its Simulink feature, the power system model along with its state feedback control is developed and simulated. The blocks used to simulate are mainly:

Transfer Function Block

Gain Block

Sum Block

Product Block

Constant Source

Scope as Sink

After mathematically obtaining the generator model and the controller models, it is developed in Simulink and then simulated. The main models used are the controller model and the synchronous machine salient pole model. The results of the simulations are as follows.

Fig 4.1 Simulink model of the power system with the DFLR

Fig 4.2 This is the output voltage. It shows how it damps the oscillation not only effectively but also very fast.

Fig 4.3 The error voltage which becomes zero finally

Fig 4.4 Simulink model of the power system with the EDFLR

Fig 4.5 This figure shows how the model of EDFLR damps the oscillations.

Fig 4.6 The error voltage which becomes zero after a while.


The DFLR and the EDFLR - the state feedback controllers have shown great potential to damp the power system oscillations efficiently and effectively. The MATLAB simulations reveal this fact. Proper tuning of these controllers is required for best results. The EDFLR can be used for multi-bus systems. In this work, certain specific model with certain approximations is used. The simulations can be tried out by taking into consideration the different models considering the different stability criteria. Hence different results will be obtained. Research work is still going on for further developments in these state feedback controllers and damping of electromechanical oscillations occurring in power system. By reducing the losses, the economy of the power utility grid is definitely benefitted.