3. intelligent frequency control

8/3/2019 3. Intelligent Frequency Control

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A DISSERTATION SUBMITTED TO THE DEPARTMENT OFELECTRON ICS AND ELECTRICAL ENGINEERING

UNIVERSITY OF STRATHCLYD E

FOR THE PART FULFILMENT OFTHE DEGREE OF MSc in

ELECTRICAL POWER ENG INEERING with BUSINESS

INTELLIGENT FREQUENCY CONTR OLON GAS TURBINE MODEL

Prepared by :HIDAYAT ZAINUDDIN

Supervised by:DR. S. JOVAN OVIC

SEPTEMBER 2005


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ABSTRACT

This project investigates the performance of knowledge-based controller toenhance the quality of generation control on gas turbine model using Simulink fromMatlab software. The proposed controller, i.e. Intelligent Turbine Controller (ITC) usesacceleration feedback to counter the over and under frequency occurrences. This type ofcontroller is also integrated with the non-fixed integral gain that represents theAutomatic G eneration Control (AGC) to minimise frequency deviations and to restorethe nominal frequency the soonest possible. Em ploying both control strategies on theconventional turbine-governor control can result in the improvement on the generationcontrol. Thus, the generator tripping and load shedding operations can be reduced andthe system can be restored back to normal condition . Finally, it is intended for a paperbased on the results obtained from this project to be published in the future.


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CHAPTER 1

INTRODUCTION

1.1 Background and Motivation

Pow er system s normally experience frequency variations. This type ofdisturbance may result from the imbalance between mechanical and electrical power ofgenerators. The power mismatch is usually caused by sudden load change. If the loadis suddenly increased, there is a deficit of generation (mechanical power less thanelectrical power) and the frequency drops. Whilst, load decrease causes an excess ofgeneration (mechanical pow er greater than electrical power) and the frequency rises.

The frequency deviations from its nominal value are used as a feedback controlsignal for the governor to control valve movem ent. In the end, the frequency willbecome stable at the new steady state level. This control schem e is referred as primaryfrequency control. In addition, the secondary frequency control provided by AutomaticGeneration Control (AGC) through integral gain control action is used to restore thefrequency to the nominal value, i.e. 50 or 60 Hz.

The magnitude of a disturbance in electrical power system s can vary from a verysmall to a very large mismatch between m echanical and electrical power. Thus, utilitieshave to provide generator and load shedding scheme to avoid damage of the generatingunits and customer load units. How ever, generator tripping as well as load shedding areundesired operations as it can reduce availability and profitability of the power systemoperation.


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This project is concerned with the design of knowledge-based controller toenhance the quality control of generator speed and power frequency. It has beenrecognised that this type of controller can reduce the over and under frequencyoccurrence [I]. On the other hand, the non-fixed integral gain can minimise thefrequency deviations and restore the nominal frequency. The project is motivated by thedesire to reduce the over and under frequency events as well as to minimise thefrequency errors and restore the nominal frequency as soon as possible. Therefore,besides the concerns on the knowledge-based controller design, the project will alsodemonstrate the effectiveness of integrating the proposed controller with non-fixedintegral gain.

1.2 Research Aims

This project aims to design an adaptive acceleration feedback controller toprevent major disturbances due to over and under frequency occurrences. The proposedcontroller will be integrated with non-fixed integral gain to minimise the frequencyerrors and restore the nominal frequency as soon as possible.

1.3 Research Approach

1.3.1 Gas Turbine M odel and Rules Design

The project involves tasks to simplify and design gas turbine model (GASTmodel) that obtained from the previous paper [2] and preparation of the rules for theproposed controller. The rules represent the control strategies to prevent over and underfrequency events, minimise frequency deviations and restore the nominal frequency.


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1.3.2 Computer Simulations

Simulink Version 6 from Matlab Version 7 [3] is used to do all the simulationtasks on the GAST model and the proposed controller. Both require relation checking,i.e. validation and verification of the GAST model and the rules from computersimulations. This is vital approach to ensure that the right system is designed(validation) and the system is built right (verification). Since, the GAST model isavailable from the previous paper, the validation is not an issue as long as theconnections are correct (verification).

Furthermore, the equivalence classes for validation and verification approach isalso being used in the project. This approach is proposed to reduce the number of testthat conducted to ensure the accuracy of the knowledge-based systems. Further detailsabout this technique will be discussed in Section 4.3.3.

1.3.3 Parameter Settings

The control strategies require parameter settings procedure for the GAST model,proposed controller and non-fixed integral gain. The existing parameters of GASTmodel will be used. However, for multi machine system, different values of speeddroop and system inertia will be used as just to represent different generator withdifferent performance.

The optimal value of controller parameters is subject to the human knowledgeand experience. Thus, sensitivity analyses on the parameters are conducted to get thedesired results from the integration of the proposed controller action and non-fixedintegral gain.


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Chapter 6 explores the visualisation of results from the GAST model itselfthrough conventional frequency control, i.e. governor speed control. This chapter alsoincludes an analysis on GAST model with integral gain only and with integral gain andproposed controller. In order to demonstrate the effectiveness of the proposed controllerseveral simulations results have been analysed including results on the multi machinesystem model.

Finally, Chapter 7 concludes in general what have been done in this project andits findings. This chapter also gives some recommendations to extend this project forfuture works.


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CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

Frequency control on a generator driven by a gas turbine is important in order toattain the system stability of power generation. A generator driven by a gas turbine canbe represented as a large rotating mass with the electro-mechanical behaviour of powersystems. Dynamic models of power system components are used in simulationprograms to simulate the electro-mechanical behaviour of gas turbine. The results ofsimulated model are used in power system planning which normally involves designingthe protection and control system. In addition, the results are also useful to designappropriate system enhancements to improve power system operation and expand thepower system network. Thus, it is vital to model the gas turbine generator accuratelysince the validity of the simulation results will depend on the parameters used formodelling the system.

The highly competitive market of electrical power industry has led to plenty ofresearch on the intelligent adaptive frequency control in improving the conventionalfrequency control actions. The conventional frequency control methods of generatingunits driven by steam or gas turbines normally include the primary control throughgovernor control action and other intelligent or non-intelligent controller that currentlybeing used in practical. The new controllers are purposely designed to enhance theperformance of the conventional controllers.


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2.2 Gas Turbine Generator Model

There are several types of gas turbine model to represent the electro-mechanicalbehaviour of gas turbine unit in generating power. One of the most comm only useddynamic model is GAST model [2, 41 (see Fig. 2.1). The GA ST model represen ts asimple open cycle, single-shaft gas turbine 141. This model is W estern Systemcoordinating Council (WSCC) compliant as it can be directly used in specificcommercial simulation programs. The G eneral Electric Company of USA recommendsthe use of GAST model for testing and modelling rather than GAST2A, which is notcomplied with WSCC requirement [2]. This is because the GAST model is moregeneral and less complicated as compared to GAST2A model that is used for studies in[5-71.

SpedRef

Fig. 2.1: GAST model to represent dynamic behaviour of gas turbine generator [2]

However, it has been recognised that the G AST model cannot give an adequaterepresentation of the temperature control loop [2]. This is explained in Fig 2.2, wherebyfor a 95 MVA gas turbine unit, the MW characteristic changes its slope when it goesbeyond 72 MW. The temperature control feedback is used when the loading levelexceeds 72 MW which is about 0.76 pu of machine rating.


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Fig. 2.2: MW versus speed reference characteristic of a 95 MVA machine [2]

Parameters of G AST model w ere calculated by previous authors in [2] by curvefitting i.e. matching the simulated and the measured responses from 13.7 MW loadrejection test since the GAST m odel parameters were not available. The authors did notmention whether it was fully or partially load rejection. There was also a single loadrejection event. As a result, we d o not know whether the parameters are really reliableand can be used for other cases in load change. But the parameters are consideredreliable since the curve discrepancy between measurement and simulation response isquite small and can be neglected as illustrated in see Fig. 2.3.

Fig. 2.3: Comparison of measurement and simulated response on 13.7 MW loadrejection on 95 MVA gas turbine [2]


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The following swing equation describes the machine speed and powerrelationship of the GAST model:

2.3 Intelligent Adaptive Controller for Frequency Control

There are several intelligent adaptive controllers for frequency control ofgenerating units to improve the performance. It has been recognised that there areresearchers who have used acceleration feedback controller by memorising the previousfrequency deviation to calculate the acceleration [l, 81 instead of directly using thefrequency deviation feedback as a control scheme [9]. The advantage of using theacceleration feedback is that the value of increasing or decreasing frequency can beexpected when acceleration and deceleration are proportional to frequency deviation.Normally, maximum frequency deviation is corresponded to the initial acceleration andvice-versa [7].

The usage of acceleration feedback has been adopted in [8] to introducestabilizing signal for each generator. The stabilizing signal is used to control theexcitation level depending on the frequency and acceleration state of the study unit. Theauthor has used the a-Af plane to prepare several simple rules for the controller. As aresult, the stabilizing signal has minimised the speed error and its ripples. However, thestabilizer did not take into account major disturbance occurrence due to over and underfrequency.

Next, knowledge-based controller for steam turbine has been proposed in [I] byintroducing acceleration feedback. The supplementary digital control from simple rulesis used to avoid over and under frequency and minimise the frequency error. The valuesof initial acceleration and deceleration are used as the controller parameters to avoidover and under frequency. The initial acceleration corresponds to maximum frequency


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deviation, while the initial deceleration is corresponded to negative maximum frequencyerror. Furthermore, current acceleration and deceleration limits have been introduced tosecure the minim al frequency errors. Thus, generator tripping and load sheddingoperation can be reduced. However, the authors did not consider the system frequencyto be restored at its nom inal value.

In [9], the author has introduced fuzzy logic based integral gain scheduling inwhich the integral gain to restore the nominal frequency is not fixed depending on thefuzzy logic membership functions. The integral gain can therefore change automaticallyin an intelligent way through fuzzy rules on the response of the generating units whenthere is a disturbance occurred. The author has adopted Sugeno technique andconventional fuzzy logic with centre of gravity defuzzification technique to change theintegral gain automatically. Thus, it is applicable to minimise the frequency error, i.e.restore the nominal frequency as soon as possible. However, this controller cannotensure in reducing generator tripping or load shedding operation when there is a majordisturbance.

2.4 Summation

It is imperative to have a set of reliable parameters for a gas turbine model torepresent its dynam ic behaviour. This is because the model can be used for computersimulations to show the results that can be expected from real gas turbine generator.GA ST model is the simplest gas turbine model that will be used in this study. Papers onintelligent controllers have given some ideas in the design of an intelligent adaptiveturbine controller for GAST model. The integration of simple rule-based controller withknowledge-based non-fixed integral gain is expected to improve the performance ofGAST model. Major frequency disturbance occurrences are expected to be reduced aswell as to restore the nom inal frequency as soon as possible.


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CHAPTER 3

FREQUENCY CONTR OL ON GAS TURBINE GENERATOR

3.1 Introduction

It is normal for a power generating plant to experience the excess or deficiencyof its generation due to the load changes, i.e. load decrease or increase in power systemnetwork. Load chang es will cause power imbalance scenario between mechanicalpower and electrical power of the generating unit. Load decrease will cause an excessof mechanical power of the gen erating unit and leads the frequ ency to rise. Whilst, loadincrease causes a deficit of mechanical power of the generation unit and frequency willdrop to compensate the power imbalance occurrence. All these are subjected to thedaily load demand within 24 hours operating time which is associated with the loadpeaks and distu rbances such as fault and loss of generation or load.

Basically, the power delivered by a generator to power system network iscontrolled by controlling the mechanical power output of a prime mover such as gasturbine, steam turbine and hydro turbine. In a gas turbine power plant, fuels are burnedto create hot gases which go through a turbine. The hot and high pressure gas spins theturbine that is coupled with generator through coupling shaft and turning the copperarmature inside the generator. This mechanism explains the power transfer frommechanical power at turbine outpu t to electrical power at genera tor output.

Thus, if the load increases, the electrical power from the generator and themechanical pow er from the gas turbine should also increase. Due to that fact, v alve


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positions regulate the fuel to be burned in order to control the mechanical power outputof gas turbine through the governor control action. Hence, if the load increases, thegovernor should signal the valve to open wider so that more fuel can be injected andburned and vice-versa.

As mentioned earlier, power changes in the power system network would alsocause the frequency deviations. After a sudden change in load, the frequency will befluctuated until it reaches a steady state level. In relation to this scenario, the generatorwill first experience changes in its speed. If the load is suddenly increased, there is adeficiency of generator and its speed and frequency will drop. Whilst, the generator willhave excess generation when the load is decreased and the generator will tend to speedup and the frequency will rise.

Therefore, the power imbalance is overcome through its effect on the generatorspeed and frequency. The frequency deviations from its nominal value are used as afeedback control signal for the governor to cause appropriate valve action. In the end,the frequency will become stable at the new steady state level. This control schemewhich provided by the governor control mechanism is referred to as primary frequencycontrol. In addition, the secondary frequency control provided by AutomaticGeneration Control (AGC) is used to restore the frequency to the nominal value, i.e. 50or 60 Hz. The frequency control of the gas turbine generator model will be discussed inthis chapter. The discussion is focused on the frequency control in isolated powersystems, i.e. generation control in an area. GAST model obtained from the previouspaper [2] has been simplified and used to represent the gas turbine-governor model as agenerating unit.

3.2 Primary Load Frequency Control

The primary load frequency control (LFC) scheme implemented by a gas turbinegenerating unit is shown in Fig. 3.1. The turbine governor control provides a negativespeed error feedback, Ao to the gas turbine through the governor. The governor adjusts


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the fuel valve positions of the gas turbine accordingly in order to compensate theimbalance between mechanical power output, P, of the gas turbine and e lectrical poweroutput, P, of the generator. In addition to this, the governo r also regulates the systemfrequency.

Pm . enerator P, & f SystemNetworkI GovernorI_

Fig. 3.1: Turbine governor control of a gas turbine generator

3.2.1 Generator Model

The generator model for GAST model is represented by swing equationexpressed in Eq. (2.1). The equation shows that the generator damping is expected toaffect the performance of the generating unit as the standard swing equation found inmany books [lo-131 does not conside r the generator damping. Therefore, the affect ofthe generator damping will be observed from the computer simulation results in Chapter6 (Section 6.2.1 and 6.2.2).

Since there is a relationship between load change and the change in frequency,Eq. (2.1) can be w ritten in the form of dev iation relationship as:

This can be expressed in Laplace transform operator notation as:


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Then, rewrite the equation to get the frequency or speed response model asfollows:

This equation can be illustrated in the form of block diagram as shown in Fig.

Fig. 3.2: Block diagram of generator model

3.2.2 Load Model

Load model that is used for the GAST model is represented by a step change ofthe electrical pow er. In Fig. 3.2, the load model, i.e. Ape is at the negative point and itrepresents a sudden load change which instantaneously causes the change in theelectrical power of the gas turbine generator. This causes a mismatch between themechanical and electrical power which in turn results in speed variations as determinedin Eq. (3.3).


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3.2.3 Turbine Model

The turbine or prime mover model for GAST model consists of two fuel systemsand a temperature control feedback. However, the temperature control feedback hasbeen eliminated in this project; thus will be further discussed in Section 3.3. In the caseof a steam turbine, the models for the prime mover must consider the steam supply andboiler system characteristic, whilst the penstock characteristics for a hydro turbine [ lo].

max I

min

Fig. 3.3: Block diagram of prime-moverlturbine modelThe prime mover model in Fig. 3.3 shows that there are two fuel system lag time

constants to represent the turbine of the GAST model as a prime mover. The first fuelsystem lag time constant, Tfl characterizes the k e l valve position time constant. Whilstthe second fuel system lag time constant, Tn describes the fuel injection before beingburned in order to produce hot gas at high pressure and high velocity that go through theturbine blades for spinning.

The model also includes the limiter due to the fact that there is a maximum andminimum limit of fuel to be injected in the combustion chamber. This will affect themechanical power output from the turbine in terms of its maximum and minimum limit.

The input of the turbine model is the change in valve position from nominalvalue, APVa,,,. The turbine model takes into account the turbine damping to obtain themechanical power output performance of the turbine. Thus, mechanical power outputfrom the prime mover is expressed by:


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3.2.4 Governor Model

The governor model of GAST model provides a mechanism that will sense thechanges in the machine speed as a result of any load change. Once it senses the speeddeviation, it will send open or close direction signal to the fuel valve to change themechanical power output to compensate the load change. The governor is modelledbased on the steady state equation of frequency-power relation for turbine governorcontrol [lo-121 as exp ressed in the following equation:

Where,AP, is change in mechan ical pow erPK f s load referenceAf is change in frequencyR is governor speed droop (HzlMW or per unit)

For synchronous machine, the electrical frequency of each generator isproportional to its rotor speed. Hence, we can replace the frequency deviation withrotational speed deviation, A o that we get from gene rator model as a feedback control inFig. 3.2. Furthermore, since we will use the output of the gove rnor as the input of theturbine model to operate the fuel valve, the change in mechanical power from governorwill be expressed as change in valve position from nominal, APvaIve. Thus, this can beexpressed as:


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The value of R from Eq. (3.5) and (3.6) determines the steady state speed versusload curve of the generating unit as shown in Fig. 3.4. The slope of the characteristicdefines the speed droop, i.e. ratio of speed or frequency deviation (Am or Af) to changein valve position or power output (AP). Thus, the governor speed droop is defined by[ l 11:

speed - r - ?equency - hange%R = x 100power - output - hange

Where,ON L is steady state speed at no loadOFL is steady state speed at full loada, is nominal or rated speed

Frequency 4

7Power output orvalve position (pu)

Fig. 3.4: Typical speed droop characteristic of a governor

For simple understanding on the governor speed droop, R=0.05 or 5% wouldmean a 5% of frequ ency deviation causing a 100%change in power output.


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The load reference, Pref is used to set reference or nominal frequency at anydesired output as illustrated in Fig. 3.5. In practical use, the load reference setting isadjusted through actuation of the speed changer motor. For 50 Hz system, if we set theload reference value of 0.5 pu, then the nominal frequency of 50 Hz will be at 0.5 puoutpu t. Therefore, the basic control input to a genera ting unit is the load reference setpoint in which a desired unit dispatch can be maintained while holding systemfrequency c lose to the desired nom inal value [I 0-1 11.

The governor model takes into consideration the governor lag time constant.This consideration is to comply the response time of the governor speed control tocompensate the sudden imbalance power between mechanical and electrical power.Typical value of the governor time constant, T, is 0.1s [12].

Load reference setting fornominal frequency at 1.0 pupower ou tput (full load)Frequency4 Load reference setting(PU) for nominal frequencyat 0.5 pu pow er output

/Load reference setting fornominal frequency at 0 pupower output (no load)1.0 Power output(Pu)

Fig. 3.5: Load reference or speed changer setting

Dead band has its own role in the governor model. The governor will send twodirections of movement to the fuel valve, i.e. open and close signal which may have avery quick response. Therefore, dead band is included at the output of the model toprevent valve to open and close at the same time when both opening and closing signals


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from governor are sent within a very short time interval [14]. This means that withoutdead band, the valve will response continuously to small change in the input signal.Consequently, it is important to have a dead band in the governor model so that thegovernor speed control will not respond to any signal that is smaller than a specified

1 minimum value [121.As a result, Fig. 3.6 describes the governor model based on Eq. (3.6) including

the consideration of governor lag time constant and dead band block.

I Dead band

Fig. 3.6: Block diagram of governor model

3.3 Turbine-Governor Control of GAST Model

The GAST model that represents the dynamic behaviour of the gas turbinegenerator is the com bination of all models that have been discussed in Section 3.2. As aresult of the combination, Fig. 3.7 forms the simplified GAST model that will be usedfor simulation in the study. In addition, by manipulating Eq. (3.3), (3.4) and (3.6), thefrequency or speed response of GAST m odel can be expressed by:

1[(ref g A u D , u r b A u

A u = 2 H x s


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Governor

dP.1- A, PITI

+Id-ea d Zone

Integ'JtOr Rotor. 1 R HDgen (H - Inertia mnd.nt)

D h l b

Fuel @em1

Generator

-uel ryrtem2I... .. ..

d h1IR - D h l b Gas Turbine

-v (GAST)1 1 1- d hT g.s+l T t l 9 1 T R r r l

GovernorP'cf -

Fig. 3.7: Simplified GAST model to represent dynamic behaviour of the gasturbine generator

The simplified GAST model is slightly different from the model previouslyshown in Fig. 2.1. There are three differences compared to the original model, i.e.addition of governor time constant block, addition of dead band and elimination oftemperature control feedback.

The governor time constant block has been added to the simplified model. TheGAST model that is used for studies in [2 , 41 as shown in Fig. 2.1 did not include thegovernor time constant. However, governor time constant is included in the GAST2Amodel which is more complicated, used for studies in [5-71. But each study useddifferent value of time constant. Hence, an assumption has been made on the value ofgovernor time constant that will be part of the model parameters. Typical value ofgovernor time constant at 0.1s will be used in all simulations. Theoretically, theaddition of the governor time constant will increase the delay time on the modelperformance. Furthermore, governor lag time constant expresses the fact that we needto take into account the response time of the governor to compensate the powerimbalance. So, all results from this project will consider governor time constant in theGAST model.

Another difference is the used of dead band in the model. It has been recognisedthat the final settling frequency of governor systems that use fuel value opening as an


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indication of mechanical power output can be effected by dead band in the turbinecontrols [2]. This disadvantage is suffered by new digital governors that use powermeasurement as input instead of rotational speed. However, the dead band will still beused in this case study but at very small threshold value which will not affect the desiredoutput of the machine. The use of dead band is due to the importance of preventingcontinuous movement caused by small change in the input signal to the fuel valve. Theimportance of dead band has been previously discussed in Section 3.2.4.

Finally, it is obvious that there is a difference in terms of the temperature controlfeedback. The temperature control feedback has been eliminated since this study doesnot apply high loading level to the gas turbine model. As discussed in Chapter 2(Section 2.2), the temperature control is used when the exhaust temperature exceeds aset temperature to avoid damaging the turbine blades but the GAST model cannot givean adequate representation of the temperature control loop [2]. This is because, for a 95MVA gas turbine unit, the temperature control is used when the loading level goesbeyond 0.76 pu of machine rating and the slope of MW characteristic is changedstarting from that point (see Fig. 2.2). Therefore, the loading level that will be used inthis project is restricted to less than 70% or 0.7 pu of machine rating.

Table 3.1: Description of GAST model parameters


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Table 3.1 gives the description of GAS T model param eters depicted in Fig. 3.7that used in this study.

3.4 Automatic Generation Control

The Automatic Generation Control (AGC) provides the secondary and tertiarycontrol. The secondary control is superimposed on the primary control that has beendiscussed previously in Section 3.2. It is important to avoid the drawback of theprimary control, i.e. steady state errors in the frequency, Af. Thus, AG C is used torestore the frequency to its nom inal value. The tertiary con trol is concerned with theeconom ic dispatch (ED) of active power by each generating unit.

Basically, the main objectives of the AGC or so called secondary frequencycontrol are to restore the frequency to its nominal value and to maintain the correctvalue of interchange power between control areas. It is also used fo r tertiary control, i.e.to ma intain ED of each generating unit.

This study focuses on the application of the AG C in isolated power systems, i.e.generation control in an area to restore the frequency to its nominal value. Thus, themaintenance of interchange power in interconnected power systems is not an issue.Moreover, the discussion will not go further into details on the ED part of eachgenerator as this study does not involve the cost curves of the generating units tocom pute the econom ic participation for each unit.

The A GC function in either 50 or 60 Hz system to restore the frequency to thespecified nominal value is accom plished by adding a reset or negative value of integralgain control, K which acts on the load reference settings of the governor model asshown in Fig. 3.8. The negative integral gain will force the frequency steady state errorto zero by adjusting the load reference set point through the speed change r motor. Thissecondary control action is much slower than the conven tional control of the governor,i.e. prim ary control.


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Fig. 3.8: GA ST m odel with integral gain ad ded at the load reference set point

3.5 Multi Machine in Isolated Power Systems

Next, the load frequency control model that has been discussed earlier will beapplied to the case of load sharing by parallel generating units in an area. For instant,we can consider system shown in Fig. 3.9. The system consists of 3-unit of gas turbinegenerator, 3-line and 3-bus with local load at each bus.

Fig. 3.9: Thre e generating units in an isolated power system

When the system experience load changes from one or more of the local loads,the total load change is described by:


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The total load change causes the power imbalance between mechanical andelectrical power of the generators which in turn the mechanical power will compensatethe mismatch. Thus, the total change in mechanical power is expressed by:

For system depicted in Fig. 3.9, the total value of generator damping, turbinedamping and system inertia are described as the following correspondingly:

It is well known that a power system has a unique value of frequency. Thismeans each generating unit has the same value of frequency or speed deviation. Thus,the speed or frequency response of GAST model for multi machine case in an area canbe expressed as:

Where,i = l , 2, 3.. ...n (n is unit of generators)

3. intelligent frequency control

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