48461536-DynCtrl-FS2010-notes-part-1

8/3/2019 48461536-DynCtrl-FS2010-notes-part-1

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Dynamics and Control of

Electric Power Systems

Lecture 227-0528-00, ITET ETH

Goran AnderssonEEH - Power Systems Laboratory

ETH Zurich

February 2010


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ii


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Contents

Preface v

1 Introduction 1

1.1 Control Theory Basics - A Review . . . . . . . . . . . . . . . 2

1.1.1 Simple Control Loop . . . . . . . . . . . . . . . . . . . 2

1.1.2 State Space Formulation . . . . . . . . . . . . . . . . . 5

1.2 Control of Electric Power Systems . . . . . . . . . . . . . . . 5

1.2.1 General considerations . . . . . . . . . . . . . . . . . . 5

2 Frequency Dynamics in Electric Power Systems 9

2.1 Dynamic Model of the System Frequency . . . . . . . . . . . 9

2.1.1 Dynamics of the Generators . . . . . . . . . . . . . . . 9

2.1.2 Frequency Dependency of the Loads . . . . . . . . . . 14

2.2 Dynamic Response of Uncontrolled Power System . . . . . . . 17

2.3 The Importance of a Constant System Frequency . . . . . . . 18

2.4 Control Structures for Frequency Control . . . . . . . . . . . 18

3 Primary Frequency Control 21

3.1 Implementation of Primary Control in the Power Plant . . . . 21

3.2 Static Characteristics of Primary Control . . . . . . . . . . . 23

3.2.1 Role of speed droop depending on type of power system 24

3.3 Dynamic Characteristics of Primary Control . . . . . . . . . . 27

3.3.1 Dynamic Model of a One-Area System . . . . . . . . . 273.3.2 Dynamic Response of the One-Area System . . . . . . 29

3.3.3 Extension to a Two-Area System . . . . . . . . . . . . 31

3.3.4 Dynamic Response of the Two-Area System . . . . . . 33

3.4 Turbine Modelling and Control . . . . . . . . . . . . . . . . . 36

3.4.1 Turbine Models . . . . . . . . . . . . . . . . . . . . . . 36

3.4.2 Steam Turbine Control Valves . . . . . . . . . . . . . 45

3.4.3 Hydro Turbine Governors . . . . . . . . . . . . . . . . 46

3.5 Dynamic Responses including Turbine Dynamics . . . . . . . 47

iii


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iv Contents

4 Load Frequency Control 51

4.1 Static Characteristics of AGC . . . . . . . . . . . . . . . . . . 514.2 Dynamic Characteristics of AGC . . . . . . . . . . . . . . . . 54

4.2.1 One-area system . . . . . . . . . . . . . . . . . . . . . 544.2.2 Two-area system unequal sizes disturbance response 564.2.3 Two-area system, unequal sizes normal control op-

eration . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.4 Two-area system equal sizes, including saturations

disturbance response . . . . . . . . . . . . . . . . . . 60

5 Synchronous Machine Model 61

5.1 Parks Transformation . . . . . . . . . . . . . . . . . . . . . . 61

5.2 The Inductance Matrices of the Synchronous Machine . . . . 655.3 Voltage Equations for the Synchronous Machine . . . . . . . . 675.4 Synchronous, Transient, and Subtransient Inductances . . . . 705.5 Time constants . . . . . . . . . . . . . . . . . . . . . . . . . . 745.6 Simplified Models of the Synchronous Machine . . . . . . . . 76

5.6.1 Derivation of the fourth-order model . . . . . . . . . . 765.6.2 The Heffron-Phillips formulation for stability studies . 79

6 Voltage Control in Power Systems 85

6.1 Relation between voltage and reactive power . . . . . . . . . 856.2 Voltage Control Mechanisms . . . . . . . . . . . . . . . . . . 876.3 Primary Voltage Control . . . . . . . . . . . . . . . . . . . . . 88

6.3.1 Synchronous Machine Excitation System and AVR . . 88

7 Stability of Power Systems 95

7.1 Damping in Power Systems . . . . . . . . . . . . . . . . . . . 957.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . 957.1.2 Causes of Damping . . . . . . . . . . . . . . . . . . . . 967.1.3 Methods to Increase Damping . . . . . . . . . . . . . . 97

7.2 Load Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 987.2.1 The Importance of the Loads for System Stability . . 987.2.2 Load Models . . . . . . . . . . . . . . . . . . . . . . . 98

References 104

A Connection between per unit and SI Units for the Swing

Equation 107

B Influence of Rotor Oscillations on the Curve Shape 109


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Preface

These lectures notes are intended to be used in the lecture Dynamics andControl of Power Systems (Systemdynamik und Leittechnik der elektrischenEnergieversorgung) (Lecture 227-0528-00, D-ITET, ETH Zurich) given atETH Zurich in the Master Programme of Electrical Engineering and Infor-mation Technology.

The main topic covered is frequency control in power systems. Theneeded models are derived and the primary and secondary frequency con-trol are studied. A detailed model of the synchronous machine, based on

Parks transformation, is also included. The excitation and voltage controlof synchronous machines are briefly described. An overview of load modelsis also given.

Zurich, February 2010

Goran Andersson

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vi Preface


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1Introduction

In this chapter a general introduction to power systems control is given.Some basic results from control theory are reviewed, and an overview of theuse of different kinds of power plants in a system is given.

The main topics of these lectures will be

Power system dynamics Power system control Security and operational efficiency.

In order to study and discuss these issues the following tools are needed

Control theory (particularly for linear systems)

Modelling Simulation Communication technology.

The studied system comprises the subsystems Electricity Generation, Trans-mission, Distribution, and Consumption (Loads), and the associated controlsystem has a hierarchic structure. This means that the control system con-sists of a number of nested control loops that control or regulate differentquantities in the system. In general the control loops on lower system levels,e.g. locally in a generator, are characterized by smaller time constants than

the control loops active on a higher system level. As an example, the Auto-matic Voltage Regulator (AVR), which regulates the voltage of the generatorterminals to the reference (set) value, responds typically in a time scale of asecond or less, while the Secondary Voltage Control, which determines thereference values of the voltage controlling devices, among which the genera-tors, operates in a time scale of tens of seconds or minutes. That means thatthese two control loops are virtually de-coupled. This is also generally truefor other controls in the systems, resulting in a number of de-coupled controlloops operating in different time scales. A schematic diagram showing thedifferent time scales is shown in Figure 1.1.

1


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2 1. Introduction

Protection

VoltageControl

TurbineControl

TieLinePowerand

FrequencyControl

1/10110100 Time(s)

Figure 1.1. Schematic diagram of different time scales of power system controls.

The overall control system is very complex, but due to the de-couplingit is in most cases possible to study the different control loops individually.This facilitates the task, and with appropriate simplifications one can quiteoften use classical standard control theory methods to analyse these con-trollers. For a more detailed analysis, one usually has to resort to computersimulations.

A characteristic of a power system is that the load, i.e. the electric powerconsumption, varies significantly over the day and over the year. This con-

sumption is normally uncontrolled. Furthermore, since substantial parts ofthe system are exposed to external disturbances, the possibility that linesetc. could be disconnected due to faults must be taken into account. Thetask of the different control systems of the power system is to keep the powersystem within acceptable operating limits such that security is maintainedand that the quality of supply, e.g. voltage magnitudes and frequency, iswithin specified limits. In addition, the system should be operated in aneconomically efficient way. This has resulted in a hierarchical control sys-tem structure as shown in Figure 1.2.

1.1 Control Theory Basics - A ReviewThe de-coupled control loops described above can be analyzed by standardmethods from the control theory. Just to refresh some of these concepts,and to explain the notation to be used, a very short review is given here.

1.1.1 Simple Control Loop

The control system in Figure 1.3 is considered. In this figure the block G(s)represents the controlled plant and also possible controllers. From this figure


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1.1. Control Theory Basics - A Review 3

SystemControl

Centre

StateEstimation

PowerFlowControl

EconomicalDispatch

Security Assessment

Generation(inpowerstation)

TurbineGovernor

VoltageGontrol

NETWORK

Power Transmission

andDistribution

TapChangerControl(DirectandQuadrature)

Reactivepowercompensation

HVDC,FACTS

Loads

Innormalcasenotcontrolled

Figure 1.2. The structure of the hierarchical control systems of a power system.

the following quantities are defined1

:

r(t) = Reference (set) value (input)

e(t) = Control error

y(t) = Controlled quantity (output)

v(t) = Disturbance

Normally the controller is designed assuming that the disturbance is equalto zero, but to verify the robustness of the controller realistic values of v

must be considered.In principle two different problems are solved in control theory:

1. Regulating problem

2. Tracking problem

1Here the quantities in the time domain are denoted by small letters, while the Laplacetransformed corresponding quantities are denoted by capital letters. In the following thisconvention is not always adhered to, but it should be clear from the context if the quantityis expressed in the time or the s domain.


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4 1. Introduction

e+

_

G(s)

H(s)

r y

v

Figure 1.3. Simple control system with control signals.

In the regulating problem, the reference value r is normally kept constantand the task is to keep the output close to the reference value even if dis-turbances occur in the system. This is the most common problem in powersystems, where the voltage, frequency, and other quantities should be keptat the desired values irrespective of load variations, line switchings, etc.

In the tracking problem the task is to control the system so that theoutput y follows the time variation of the input r as good as possible. Thisis sometimes also called the servo problem.

The transfer function from the input, R, to the output, Y, is given by

(in Laplace transformed quantities)

F(s) =Y(s)

R(s)=

C(s)

R(s)=

G(s)

1 + G(s)H(s)(1.1)

In many applications one is not primarily interested in the detailed timeresponse of a quantity after a disturbance, but rather the value directly afterthe disturbance or the stationary value when all transients have decayed.Then the two following properties of the Laplace transform are important:

g(t 0+) = lims

sG(s) (1.2)

andg(t ) = lim

s0sG(s) (1.3)

where G is the Laplace transform of g. If the input is a step function,Laplace transform = 1/s, and F(s) is the transfer function, the initial andstationary response of the output would be

y(t 0+) = lims

F(s) (1.4)

andy(t ) = lim

s0F(s) (1.5)


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1.2. Control of Electric Power Systems 5

1.1.2 State Space Formulation

A linear and time-invariant controlled system is defined by the equations

x = Ax + Bu

y = Cx + Du

(1.6)

The vector x = (x1 x2 . . . xn)T contains the states of the system, which

uniquely describe the system. The vector u has the inputs as components,and the vector y contains the outputs as components. The matrix A, ofdimension n n, is the system matrix of the uncontrolled system. Thematrices B, C, and D depend on the design of the controller and the availableoutputs. In most realistic cases D = 0, which means that there is zerofeedthrough, and the system is said to be strictly proper. The matrices Aand B define which states are controllable, and the matrices A and C definewhich states are observable. A controller using the outputs as feedbacksignals can be written as u = Ky = KCx, assuming D = 0, where thematrix K defines the feedback control, the controlled system becomes

x = (A BK C)x (1.7)

1.2 Control of Electric Power Systems

1.2.1 General considerations

The overall control task in an electric power system is to maintain the bal-ance between the electric power produced by the generators and the powerconsumed by the loads, including the network losses, at all time instants.If this balance is not kept, this will lead to frequency deviations that if toolarge will have serious impacts on the system operation. A complication isthat the electric power consumption varies both in the short and in the longtime scales. In the long time scale, over the year, the peak loads of a day arein countries with cold and dark winters higher in the winter, so called winterpeak, while countries with very hot summers usually have their peak loadsin summer time, summer peak. Examples of the former are most European

countries, and of the latter Western and Southern USA. The consumptionvaries also over the day as shown in Figure 1.4. Also in the short run theload fluctuates around the slower variations shown in Figure 1.4, so calledspontaneous load variations.

In addition to keeping the above mentioned balance, the delivered elec-tricity must conform to certain quality criteria. This means that the voltagemagnitude, frequency, and wave shape must be controlled within specifiedlimits.

If a change in the load occurs, this is in the first step compensated bythe kinetic energy stored in the rotating parts, rotor and turbines, of the


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6 1. Introduction

holidays

0

20

40

60

80

100

1 4 7 10 13 16 19 22

0

20

40

60

80

100

120

1 4 7 10 13 16 19 22

weekdays

holidaysholidays

weekdays

Time Time

Loa

d(%)

Loa

d(%)

Figure 1.4. Typical load variations over a day. Left: Commercialload.; Right: Residential load.

generators resulting in a frequency change. If this frequency change is toolarge, the power supplied from the generators must be changed, which is donethrough the frequency control of the generators in operation. An unbalancein the generated and consumed power could also occur as a consequence ofthat a generating unit is tripped due to a fault. The task of the frequencycontrol is to keep the frequency deviations within acceptable limits during

these events.To cope with the larger variations over the day and over the year gener-ating units must be switched in and off according to needs. Plans regardingwhich units should be on line during a day are done beforehand based onload forecasts2. Such a plan is called unit commitment. When making sucha plan, economic factors are essential, but also the time it takes to bring agenerator on-line from a state of standstill. For hydro units and gas tur-bines this time is typically of the order of some minutes, while for thermalpower plants, conventional or nuclear, it usually takes several hours to getthe unit operational. This has an impact on the unit commitment and onthe planning of reserves in the system3.

Depending on how fast power plants can be dispatched, they are clas-sified as peak load, intermediate load, or base load power plants. Thisclassification is based on the time it takes to activate the plants and on the

2With the methods available today one can make a load forecast a day ahead whichnormally has an error that is less than a few percent.

3In a system where only one company is responsible for the power generation, the unitcommitment was made in such a way that the generating costs were minimized. If severalpower producers are competing on the market, liberalized electricity market, the situationis more complex. The competing companies are then bidding into different markets, pool,bi-lateral, etc, and a simple cost minimizing strategy could not be applied. But also inthese cases a unit commitment must be made, but according to other principles.


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1.2. Control of Electric Power Systems 7

fuel costs and is usually done as below4. The classification is not unique and

might vary slightly from system to system.

Peak load units, operational time 10002000 h/a Hydro power plants with storage

Pumped storage hydro power plants

Gas turbine power plants

Intermediate load units, operational time 30004000 h/a Fossil fuel thermal power plants

Bio mass thermal power plants

Base load units, operational time 50006000 h/a Run of river hydro power plants

Nuclear power plants

In Figure 1.5 the use of different power plants is shown in a load durationcurve representing one years operation.

The overall goal of the unit commitment and the economic dispatch isthe

Minimization of costs over the year

Minimization of fuel costs and start/stop costs

4The fuel costs should here be interpreted more as the value of the fuel. For a hydropower plant the fuel has of course no cost per se. But if the hydro plant has a storagewith limited capacity, it is obvious that the power plant should be used during high loadconditions when generating capacity is scarce. This means that the water value is high,which can be interpreted as a high fuel cost.


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8 1. Introduction

System Load

Time1 year

Run of river hydro power

Nuclear power

Controllable hydro power

Thermal power (fossil fuel)

Hydro power reserves

Gas turbines

Figure 1.5. Duration curve showing the use of different kinds of power plants.


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2Frequency Dynamics in Electric Power

Systems

In this chapter, the basic dynamic frequency model for a large power systemis introduced. It is based on the swing equation for the set of synchronousmachines in the system. Certain simplifications lead to a description of thedominant frequency dynamics by only one differential equation which can

be used for the design of controllers. Also of interest is the frequency de-pendency of the load in the system, which has a stabilizing effect on the

frequency. Note that no control equipment is present yet in the models pre-sented in this chapter: the system is shown in open loop in order to un-derstand the principal dynamic behaviour. Control methods are presented inthe subsequent chapters.

2.1 Dynamic Model of the System Frequency

In order to design a frequency control methodology for power systems, theelementary dynamic characteristics of the system frequency have to be un-derstood. For this purpose, a simplified model of a power system with sev-eral generators (synchronous machines) will be derived in the sequel. Thenominal frequency is assumed to be 50 Hz as in the ENTSO-E Continen-tal Europe system (former UCTE). Generally, deviations from this desiredvalue arise due to imbalances between the instantaneous generation and con-sumption of electric power, which has an accelerating or decelerating effecton the synchronous machines.

2.1.1 Dynamics of the Generators

After a disturbance in the system, like a loss of generation, the frequencyin different parts of a large power system will vary similar to the exemplaryillustration shown in Figure 2.1. The frequencies of the different machinescan be regarded as comparatively small variations over an average frequencyin the system. This average frequency, called the system frequency, is thefrequency that can be defined for the socalled centre of inertia (COI) of thesystem.

We want to derive a model that is valid for reasonable frequency devia-tions. For this purpose, the exact version of the swing equation will be used

9


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10 2. Frequency Dynamics in Electric Power Systems

50

49.8

49.6

49.4

49

48.8

49.2

0 1 2 3 4 5

f(Hz)

t(s)

Figure 2.1. The frequency in different locations in an electric powersystem after a disturbance. The thicker solid curve indicates the aver-age system frequency. Other curves depict the frequency of individualgenerators.

to describe the dynamic behaviour of generator i:

i =0

2Hi(Tmi(p.u.) Tei(p.u.)) , (2.1)

with the usual notation. The indices m and e denote mechanical (turbine)and electrical quantities respectively. i is the absolute value of the rotorangular frequency of generator i. The initial condition for eq. (2.1), thepre-disturbance frequency, is normally the nominal frequency i(t0) = 0.Of main interest is usually the angular frequency deviation i:

i = i 0 . (2.2)By deriving eq. (2.2) with respect to the time, one obtains i = i. Note

that for rotor oscillations the frequency of i is often of interest, while theamplitude of i is the main concern in frequency control. Also note thatfor the initial condition i(t0) = 0 holds if eq. (2.1) is formulated for .

In order to convert the torques in eq. (2.1) to power values, the relationP(p.u.) = T(p.u.)

0is used, which yields:

i =20

2Hii(Pmi(p.u.) Pei(p.u.)) , (2.3)

The power can also be expressed in SIunits (e.g. in MW instead of p.u.) bymultiplication with the power base SBi, which represents the rated power of


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2.1. Dynamic Model of the System Frequency 11

GT

eP

Load

lo a dP

Generator

Turbine

set

mP

mP

Figure 2.2. Simplified representation of a power system consisting ofa single generator connected to the same bus as the load.

the generator i. Furthermore, eq. (2.3) can be rewritten such that the uniton both sides is MW:

2HiSBi

0

i =0

i

(Pmi

Pei) . (2.4)

Note that this is still the exact version of the swing equation, which isnonlinear. For further details on different formulations of the swing equa-tion, please refer to Appendix A. Now, the goal is to derive the differentialequation for the entire system containing n generators. In a highly meshedsystem, all units can be assumed to be connected to the same bus, represent-ing the centre of inertia of the system. With further simplifications, theycan even be condensed into one single unit. An illustration of this modellingis depicted in Figure 2.2. A summation of all the equations (2.4) for the n

generators in the system yields

2n

i=1

HiSBi1

0 i =

ni=1

0i

(Pmi Pei) . (2.5)

Because of the strong coupling of the generation units, i = can be as-


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+ */*

2

0

2 BHS

1

s0

eP

mP

a /a b

b

Figure 2.3. Block diagram of nonlinear frequency dynamics as in eq. (2.11).

sumed for all i. By defining the quantities

=

i Hi i

HiCentre of Inertia frequency (2.6)

SB =i

SBi Total rating, (2.7)

H =

i HiSBii SBi

Total inertia constant, (2.8)

Pm =i

Pmi Total mechanical power, (2.9)

Pe = i

Pei Total electrical power, (2.10)

the principal frequency dynamics of the system can be described by thenonlinear differential equation

=20

2HSB(Pm Pe) . (2.11)

Eq. (2.11) is illustrated as a block diagram in Figure 2.3. For the frequency in the centre of inertia holds as well

= 0 + . (2.12)

In order to obtain a linear approximation of eq. (2.11), = 0 can beassumed for the right-hand side. This is a valid assumption for realisticfrequency deviations in power systems. This yields

=0

2HSB(Pm Pe) . (2.13)

The dynamics can also be expressed in terms of frequency instead of angularfrequency. Because of = 2f and = 2f follows

f =f0

2HSB(Pm Pe) . (2.14)


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A very simple and useful model can be derived if some more assumptions

are made. The overall goal of our analysis is to derive an expression thatgives the variation of after a disturbance of the balance between Pm andPe. Therefore, we define

Pm =i

Pmi = Pm0 + Pm , (2.15)

where Pm0 denotes the mechanical power produced by the generators insteady state and Pm denotes a deviation from that value. The total gen-erated power is consumed by the loads and the transmission system losses,i.e.

Pe =i

Pei = Pload + Ploss , (2.16)

which can, in the same way as in eq. (2.15), be written as

Pe = Pe0 + Pload + Ploss (2.17)

with

Pe0 = Pload0 + Ploss0 . (2.18)

If the system is in equilibrium prior to the disturbance,

Pm0 = Pe0 (2.19)

and

Pm0 = Pload0 + Ploss0 (2.20)

are valid. Furthermore, the transmission losses after and before the distur-bance are assumed to be equal, i.e.

Ploss = 0 . (2.21)

If neither the disturbance nor the oscillations in the transmission system are

too large, these approximations are reasonable. Using eqs. (2.15) (2.21),eq. (2.13) can now be written as

=0

2HSB(Pm Pload) , (2.22)

or equivalently

f =f0

2HSB(Pm Pload) . (2.23)

Eq. (2.23) can be represented by the block diagram in Figure 2.4.


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f

load

P

mP

System inertia

0

(2 )B

f

HS s

Figure 2.4. Linearized model of the power system frequency dynamics.

2.1.2 Frequency Dependency of the Loads

Loads are either frequency-dependent or frequency-independent. In realpower systems, a frequency dependency of the aggregated system load isclearly observable. This has a stabilizing effect on the system frequency f,as will be shown in the sequel. Apart from a component depending directlyon f, large rotating motor loads cause an additional contribution dependingon f. This is due to the fact that kinetic energy can be stored in the rotatingmasses of the motors.

A load model that captures both effects is given by

Pfload Pf0load = Pfload = Klf + g(f) (2.24)

where

Pf0load

: Load power when f = f0,

Kl: Frequency dependency,

g(f): Function that models the loads with rotating masses.

The function g(f) will now be derived. The rotating masses have thefollowing kinetic energy:

W(f) = 12

J(2f)2 (2.25)

The change in the kinetic energy, which is equal to the power PM consumedby the motor, is given by

PM =dW

dt(2.26)

and

PM =dW

dt. (2.27)


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W can be approximated by

W(f0 + f) = 22J(f0 + f)

2 =

W0 + W = 22Jf20 + 2

2J2f0f + 22J(f)2

= W0 +2W0

f0f +

W0f20

(f)2

W 2W0f0

f

PM 2W0f0

df

dt=

2W0f0

f (2.28)

The frequency dependency of the remaining load can also be written as

Ploadf

f = Klf =1

Dlf . (2.29)

The values of W0 and Dl are obviously highly dependent on the structureof the load and can be variable over time. Especially W0 is only a factorin power systems with large industrial consumers running heavy rotatingmachines. The constant Dl has typical values such that the variation of theload is equal to 0 . . . 2 % per % of frequency variation.

The block diagram in Figure 2.5 represents the dynamic load model. To-gether with the power system dynamics derived before, we obtain a dynam-ical system with a proportional/differential control caused by the loads.

However, this effect is too small to be able to keep the frequency within rea-sonable bounds. As we will see in the next section, the absence of any othercontrol equipment would lead to unacceptable and remaining frequency de-viations even for moderate disturbances.

The power system model derived so far is shown in Figure 2.6.


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f

f

loadP

1

lD

0

0

2Ws

f

Figure 2.5. Block diagram of the dynamic load model.

loadP

f

eP

mP

loadP

0

0

2Ws

f

Rotating mass loads

Frequency-dependent loads

Power generation change

System inertia

System load change

1

lD

0

(2 )B

f

HS s

Figure 2.6. Model of power system without control.


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2.2. Dynamic Response of Uncontrolled Power System 17

2.2 Dynamic Response of Uncontrolled Power System

Now we will conduct a numerical simulation of the uncontrolled frequencydynamics after a disturbance. Both loss of generation and loss of load willbe shown, represented by a positive resp. negative step input on the variablePload. Table 2.1 displays the parameters used in the simulation. In Figure2.7, a time plot of the system frequency is shown corresponding to thedifferent disturbances. Note that this result is purely theoretic as such largefrequency deviations could never occur in a real power system because ofvarious protection mechanisms. However, it illustrates well the possiblefrequency rise or decay and the stabilizing self-control effect caused by thefrequency dependency of the load.

Parameter Value Unit

H 5 sSB 10 GWf0 50 Hz

DL1

200

Hz

MW

W0 100MW

Hz

Table 2.1. Parameters for time domain simulation of power system.

The plot shows the time evolution of the system frequency for a dis-turbance of (top to bottom) Pload =

1000 MW, Pload =

500 MW,

Pload = 100 MW (sudden loss of load) and Pload = 100 MW, Pload =500 MW, Pload = 1000 MW (sudden increase of load or loss of generation).

0 10 20 30 40 50 6044

46

48

50

52

54

56

Time [s]

Sys

temFrequency[Hz]

1000 MW

500 MW

100 MW

+100 MW

+500 MW

+1000 MW

Figure 2.7. Theoretical frequency responses of uncontrolled powersystem (DL = 1/200 Hz/MW).


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2.3 The Importance of a Constant System Frequency

In the most common case Pm Pload is negative after a disturbance, likethe tripping of a generator. It is also p ossible that the frequency rises dur-ing a disturbance, for example when an area that contains much generationcapacity is isolated. Since too large frequency deviations in a system arenot acceptable, automatic frequency control, which has the goal of keepingthe frequency during disturbances at an acceptable level, is used. Further-more, the spontaneous load variations in an electric power system result ina minutetominute variation of up to 2%. This alone requires that someform of frequency control must be used in most systems.

There are at least two reasons against allowing the frequency to deviate

too much from its nominal value. A nonnominal frequency in the systemresults in a lower quality of the delivered electrical energy. Many of thedevices that are connected to the system work best at nominal frequency.Further, too low frequencies (lower than 47 48 Hz) lead to damagingvibrations in steam turbines, which in the worst case have to be discon-nected. This constitutes an even worse stress on the system and can lead toa complete power system collapse. In comparison with thermal units, hydropower plants are more robust and can normally cope with frequencies downto 45 Hz.

2.4 Control Structures for Frequency Control

In the following two chapters, the control structures that ensure a constantsystem frequency of 50 Hz will be described. The automatic control systemconsists of two main parts, the primary and secondary control. Tertiarycontrol, which is manually activated in order to release the used primary andsecondary control reserves after a disturbance, is not discussed here. This isdue to the fact that the utilization of tertiary control reserves is more similarto the electricity production according to generations schedules (dispatch),which are based on economic off-line optimizations.

The primary control refers to control actions that are done locally (on

the power plant level) based on the setpoints for frequency and power. Theactual values of these can be measured locally, and deviations from the setvalues results in a signal that will influence the valves, gates, servos, etc. ina primary-controlled power plant, such that the desired active power outputis delivered. In primary frequency control, the control task of priority isto bring the frequency back to (short term) acceptable values. However,there remains an unavoidable frequency control error because the controllaw is purely proportional. The control task is shared by all generatorsparticipating in the primary frequency control irrespective of the location ofthe disturbance. Further explanations follow in chapter 3.


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2.4. Control Structures for Frequency Control 19

In the secondary frequency control, also called Load Frequency Control,

the power setpoints of the generators are adjusted in order to compensate forthe remaining frequency error after the primary control has acted. Apartfrom that, another undesired effect has to be compensated by secondarycontrol: active power imbalances and primary control actions cause changesin the load flows on the tie-lines to other areas, i.e. power exchanges notaccording to the scheduled transfers. The secondary control ensures bya special mechanism that this is remedied after a short period of time.Note that in this control loop the location of the disturbance is consideredwhen the control action is determined: only disturbances within its owncontrol zone (area) are seen by the secondary controller. Note that LoadFrequency Control can also be performed manually as in the Nordel power

system. In the ENTSO-E Continental Europe interconnected system, anautomatic scheme is used, which can also be called Automatic GenerationControl. This is further discussed in chapter 4.

The basic control structures described above are depicted in Figure 2.8.Figure 2.9 shows an illustration of the time spans in which these differentcontrol loops are active after a disturbance. Note that primary and sec-ondary control are continuously active also in normal operation of the gridin order to compensate for small fluctuations. Conversely, the deploymentof tertiary reserves occurs less often.

Under-frequency load shedding is a form of system protection and acts ontimescales well under one second. As the activation of this scheme implies

the loss of load in entire regions, it must only be activated if absolutelynecessary in order to save the system. In the ENTSO-E Continental Europesystem, the first load shedding stage is activated at a frequency of 49 Hz,causing the shedding of about 15 % of the overall load. In many systems,a rotating scheme for how the load should be shed, if that is necessary, isdevised. Such a scheme is often called rotating load shedding.


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Turbine

Governor

Valves or

GatesTurbine Generator

Electrical system

Loads

Transmission lines

Other generators

Speed

Automatic

Generation

Control (AGC)

(Secondary

control loop)

Setpoint

Calculation

Power,

Frequency

Tie-Line Powers

(Internal turbine

control loop)

(Primary control loop)

Setpoint from

power generation

schedule and

tertiary control

identical for synchronous

machines in steady state

System Frequency

Turbine

Controller

Load Shedding

(emergency control)

Figure 2.8. Basic structure of frequency control in electric power systems.

30 s 15 min 60 min

Primary

Control

Secondary

Control

Tertiary

Control

Generation

Rescheduling

Power

Time

Figure 2.9. Temporal structure of control reserve usage after a dis-turbance (terminology according to ENTSO-E for region ContinentalEurope (former UCTE).


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3Primary Frequency Control

In this chapter, the mechanisms for primary frequency control are illustrated.As will be shown, they are implemented entirely on the power plant level. Inan interconnected power system, not all generation units need to have pri-mary control equipment. Instead, the total amount of necessary primarycontrol reserves are determined by statistical considerations and the distri-bution on the power plants can vary. Primary control can also be used in

islanded operation of a single generator. These different applications will bediscussed along with their principal static and dynamic characteristics.

3.1 Implementation of Primary Control in the Power

Plant

For a thermal unit, a schematic drawing of the primary control is shown inFigure 3.1. The turbine governor (depicted here together with the internalturbine controller) acts on a servomotor in order to adjust the valve through

which the live steam (coming from the boiler with high pressure and hightemperature) flows to the turbines. In the high pressure turbine, part of theenergy of the steam is converted into mechanical energy. Often the steam isthen reheated before it is injected into a medium pressure or low pressureturbine, where more energy is extracted from the steam. In practice, theseturbine-generator systems can be very large. In a big thermal unit of rating1000 MW, the total length of the turbine-generator shaft may exceed 50 m.

The control law, which is shown as a block diagram in Figure 3.2, is aproportional feedback control. It establishes an affine relation between themeasured frequency and the power generation of the plant in steady state.

Note that the turbine dynamics plays in important role in the overall

dynamical response of the system. However, we will neglect this for thesake of simplicity in the first three sections of this chapter. In section 3.4.1,different steam and hydro turbines and their modelling are discussed andlater their effect on the dynamical response will be shown.

Primary control is implemented on a purely local level; there is no co-ordination between the different units. This is also why the control lawcannot have an integral component: integrators of different power plantscould start competing each other for power production shares, which canlead to an unpredictable and unreasonable distribution of power generationon the available plants.

21


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22 3. Primary Frequency Control

Measured values

LP

Electricity gridElectrical

power

Mech.

power

HP

Actuation

Steam

Steam

Control

signals

Reference values

GBoiler

Valve

ServomotorController/Governor

Reheater

0 0,f P

,f P

Figure 3.1. Schematic drawing of the primary control installed ina thermal unit. HP = High Pressure Turbine. LP = Low Pressureturbine.

GT

Proportionalcontrol law

1/K S 0setf ffset

mP0

set

mP

f, set totm

P

Internal

Turbine

Controller

,set tot

m mP P

eP

Turbine shaft

Figure 3.2. Block diagram describing the primary control law. Themeasured power value corresponds to Pm in our notation, while inpractice the measurement can be done on the electrical side.


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3.2. Static Characteristics of Primary Control 23

3.2 Static Characteristics of Primary Control

First, it is of particular interest to study the properties of the primary fre-quency control in steady state. Referring to the block diagram in Figure 3.2,the equation describing the primary control is given by

(f0 f) 1

S+ (Psetm0 Pset,totm ) = 0 , (3.1)

which can be written as

S = f0 fPsetm0 Pset,totm

= f f0Pset,totm Psetm0

Hz/MW > 0 (3.2)

or in per unit as

S = f f0

f0

Pset,totm Psetm0Psetm0

. (3.3)

Under the assumption that the turbine power controller has an integratingcharacteristic ( 0 when t in Figure 3.2), it follows that in steadystate holds Pm = P

set,totm .

The speed droop characteristic, Figure 3.3, yields all possible steady-state operating points (Pset,totm , f) of the turbine. The position and slopeof the straight line can be fixed by the parameters Psetm0 , f0 and S. Wehave chosen to label the horizontal axis with the power Pset,totm which forsmall deviations of the frequency around the nominal value is identical tothe torque T. In the literature the speed droop characteristics is sometimesalso described by instead of by f.

0 ( . .) f p u

,( . .)

set tot

m P p u0 ( . .)

set

m P p u

( . .) f p u

( . .)S p u

Figure 3.3. Static characteristic of primary control.


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3.2.1 Role of speed droop depending on type of power system

We will now study how the frequency control of a generator will act inthree different situations. First, when the generator is part of a large inter-connected system, and second when the generator is in islanded operationfeeding a load. The third system to be studied is a two machine system.

Generator in Large System If a generator is embedded in a large inter-connected system, it can be modelled with a very good approximation asconnected to an infinite bus. This is shown in Figure 3.4.

In steady state the frequency is given by the grid frequency fG (repre-sented by the infinite bus). From the speed droop characteristics, Figure 3.5,the power produced by the generator can then be determined. The turbinegovernor controls thus only the power, not the frequency, see Figure 3.5.

0 0,set

mf P

,Gf P

LXePmPT

Infinite

Bus

Gf f

GU U

GR

Figure 3.4. Generator operating in a large interconnected system.

f

0

set

mP

0f S

,( )

set tot

m GP f

0 0und are set in the generatorset

mf P

( )G P g f

, set tot

mP

Gf

Figure 3.5. Speed droop characteristics for the case when the gener-ator is connected to an infinite bus (large system).


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3.2. Static Characteristics of Primary Control 25

Islanded Operation As depicted in Figure 3.6, the generator feeds a load in

islanded operation, which here is assumed to be a purely resistive load. Bya voltage controller the voltage U is kept constant and thus also Pe. In thiscase the primary control loop will control the frequency, not the power. Theresulting frequency can be determined from the speed droop characteristics,Figure 3.7.

0 0,set

mf P

,f P

ePmPT

U R

2

.e

U P const

R

GR

Voltage Control

Figure 3.6. Generator in islanded operation.

f

0

set

mP

0f S

,set tot

m eP P

0 0und are set in the generatorset

mf P

( ) f h P

, set tot

mP

f

Figure 3.7. Speed droop characteristics for the case when the gener-ator is in islanded operation.

Two Generator System The two generator system, Figure 3.8, provides asimple model that is often used to study the interaction between two areasin a large system. In this model the two generators could represent twosubsystems, and the speed droop is then the sum of all the individual speeddroops of the generators in the two subsystems, Figure 3.9. With the help ofthe speed droop characteristics of the two systems, we will determine how achange in load will be compensated by the two systems. Thus, if we have achange Pload of the overall load, what will the changes in Pm,1, Pm,2, andf be?


32/119


This will be solved in the following way:

The quantities (Psetm0,1, f0,1, S1) and (Psetm0,2, f0,2, S2) describe the speeddroop characteristics of the two systems g1 and g2.

From these the sum g3 = g1 + g2 is formed. From the given Pload we can determine Psetm,1, Psetm,2 and fN from g3. In a similar way: From Pload+Pload can Psetm0,1+Psetm,1, Psetm0,2+Psetm,2

and fN + f be determined, and thus Psetm,1, P

setm,2 und f.

All these steps are shown in Figure 3.9.

G2G1

,1 ,2 atload m m N P P P f

0,1 0,1 1, ,setm P f S 0,2 0,2 2, ,set

m P f S

,1mP ,2mP

,1mP

,2mP

Figure 3.8. Two generator system.

1 1, gS

2 2, gS

3 3, gS

0,2f

0,1f

Nf f

0,2

set

mP0,1set

mP

,1

set

mP ,2set

mP loadP

load load P P ,1 ,1

set set

m mP P ,2 ,2 set set

m mP P

Nf

Figure 3.9. Speed droop characteristics for a two machine system.


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3.3. Dynamic Characteristics of Primary Control 27

3.3 Dynamic Characteristics of Primary Control

3.3.1 Dynamic Model of a One-Area System

In this section, we are going to extend the dynamic frequency model intro-duced in Chapter 2 by the primary control loop in the power plants. For anindividual generator, the block diagram has been introduced in the previoussection.

Following the nomenclature introduced in Figure 3.2, we start with

Pset,totm = Psetm0 + P

setm = P

setm0

1

Sf , (3.4)

where Pset,totm describes the power setpoint of the turbine including the sched-

uled value Psetm0 and the component imposed by primary control. For the

linearized consideration of the power system in quantities, we set

Psetm0 = Psetm0 (3.5)

as the steady-state component cancels out against the other steady-statequantities. The remaining question is how a change in Pset,totm translatesinto an actual mechanical power output change Pm. Thus, we regard nowthe internal turbine control loop as depicted in Figure 3.10.

Turbine

( )tG s

Turbine

Controller

tKs

set

mP

mP

0

set

mP

Figure 3.10. Block diagram of the turbine and turbine control dynamics.

From this figure it follows that

Pm(s) =Gt(s)

Gt(s) +

1

Kts

Psetm0(s) + Psetm (s)

. (3.6)

If the dynamics of the turbine is neglected (Gt(s) = 1), one obtains

Pm(s) =1

1 + Tts

Psetm0 (s) + P

setm (s)

. (3.7)

Note that the time constant Tt = 1/Kt is fairly small compared with thefrequency dynamics of the system. Regarding now only the frequency controlcomponent (Psetm0 = 0), we obtain in steady state as before

Pm = 1S

f . (3.8)


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For the case of n controllers for n generators we obtain analogously for

controller iPmi =

1

Sif i = 1, . . . , n (3.9)

i

Pmi = i

1

Sif (3.10)

wherePm =

i

Pmi (3.11)

is the total change in turbine power. By defining

1

S =i

1

Si (3.12)

we thus have

Pm = 1

Sf . (3.13)

This can be inserted into the dynamic system frequency model derived inChapter 2 as depicted in Figure 3.11.

loadP

f

eP

loadP

0

0

2Ws

f

Rotating mass loads


System inertia

System load change1

lD

0

(2 )B

f

HS s

mP

1

1tT s

set

mP

0

set

mP

1

S

Turbine

dynamics/

control

Primary

control

Figure 3.11. Dynamic frequency model of the power system withprimary-controlled power plants.


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3.3.2 Dynamic Response of the One-Area System

Now we are going to study the effect of a disturbance in the system derivedabove. Both loss of generation and loss of load can be simulated by imposinga positive or negative step input on the variable Pload. A change of the setvalue of the system frequency f0 is not considered as this is not meaningfulin real power systems.

From the block diagram in Figure 3.11 it is straightforward to derive thetransfer function between Pload and f (P

setm0 = 0):

f(s) = 1 + sTt1

S+

1

Dl(1 + sTt) + (

2W0f0

+2HSB

f0)s(1 + sTt)

Pload(s) (3.14)

The step response for

Pload(s) =Pload

s(3.15)

is given in Figure 3.12. The frequency deviation in steady state is

f = lims0

(s f(s)) = Pload1

S+

1

Dl

=Pload

1

DR

= Pload DR (3.16)

with1

DR

=1

S

+1

Dl

(3.17)

In order to calculate an equivalent time constant Teq, Tt is put to 0. Thiscan be done since for realistic systems the turbine controller time constantTt is much smaller than the time constant of the frequency dynamics TM:

Tt TM =f0SB

(2W0

f0+

2HSBf0

) . (3.18)

This means that the transfer function in eq. (3.14) can be approximated bya first order function

f(s) =Pload(s)

1DR+ TMSBf0

s

=1

1 + TMDR SBf0s

DRPload

s

(3.19)

or

f(s) =1

1 + TMDRSBf0

s

fs

(3.20)

with

Teq = TMDRSBf0

(3.21)

as the equivalent time constant.


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0 1 2 3 4 5 6 7 8 9 1049.75

49.8

49.85

49.9

49.95

50

SystemFrequency[Hz

]

0 1 2 3 4 5 6 7 8 9 10

0

100

200

300

400

Time [s]

Power[MW

]

Pload

Pm

Pload

f

Figure 3.12. Behaviour of the one-area system (no turbine dynamics,parameterized as described in the example below) after a step increasein load. The upper plot shows the system frequency f. The lower plot

shows the step function in Pload, the increase in turbine power Pm,and the frequency-dependent load variation Pfload.

Example

SB = 4000 MW f0 = 50 Hz S = 4% = 0.04 f0

SB= 0.0450

4000Hz/MW = 1

2000Hz/MW

Dl = 1 %/1 % Dl = f0SB =50

4000Hz/MW = 1

80Hz/MW

Pload = 400 MW TM = 10 s

Then follows

f = DR Pload = 1

2000MWHz

+ 80MWHz

400M W = 0.19 Hz (3.22)

and

Teq = 10 s 1

2000MWHz

+ 80MWHz

4000 MW

50 Hz= 0.39 s . (3.23)


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3.3.3 Extension to a Two-Area System

Up to now we have mostly studied the behaviour of a power system consist-ing of a single area. If the power system is highly meshed in this case, it canbe represented by a single bus where all units are connected. In practice,however, a large interconnected power system is always divided into variouscontrol zones or areas, corresponding e.g. to countries. Understandingthe interactions between these areas is therefore highly important for theflawless operation of the entire system. For simplified simulation studies, asystem with two areas can be represented by two single bus systems with atie-line in between them. This is depicted in Figure 3.13.

G1

0,1

set

mPT1

1f

,1eP

Load 1

,1lo a dP

1 2TP

Tie-line

power flow

G2 T2

2f

,1mP

, 2eP , 2mP

0,2

set

mP

Load 2

, 2lo a dP

Area 1 Area 2

R1 R2

Figure 3.13. Simplified representation of a power system with two areas.

In order to adapt our dynamic frequency model accordingly, the powerexchange PT12 over the tie line between the areas 1 and 2 has to be modelled.This is given by

PT12 =U1U2

Xsin(1 2) (3.24)

where X is the (equivalent) reactance of the tie line. For small deviations(U1 and U2 are constant) one gets

PT12 = PT121

1 + PT122

2 = U1U2X

cos(0,1 0,2)(1 2)(3.25)

orPT12 = PT(1 2) (3.26)

with

PT =U1U2

Xcos(0,1 0,2) . (3.27)

By using this model, the block diagram of the power system can be extendedas shown in Figure 3.14.


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,1loadP

1f

,1eP

,1

f

loadP

,2loadP

2f

,2eP

,2

f

loadP

12TP

12TP12TP

0,1

0

2Ws

f

Rotating mass loads


System inertia

,1

1

lD

0

1 ,1(2 )B

f

H S s,1mP

,1

1

1tT s

,1

set

mP

0,1

set

mP

1

1

S

Turbine

dynamics/

control

Primary

control

0,2

0

2Ws

f

Rotating mass loads


System inertia

,2

1lD

0

2 ,2(2 )

B

f

H S s

,2mP

1

1t

T s

,2

set

mP

0,2

set

mP

2

1

S

Turbinedynamics/

control

Primary

control

2 TP

s

2 TP

s

Figure 3.14. Two-area dynamic model including tie-line flows.


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3.3.4 Dynamic Response of the Two-Area System

For the case regarded here, it is assumed that one of the areas is much smallerthan the other. The bigger one of the two areas can then be regarded as aninfinite bus in our analysis. We will now study the behaviour after a loadchange in the smaller area, Area 1. The system to be studied is depicted inFigure 3.15.

12TP

X1 1,U 2 2,U

1

2 0f f

Area 1

1 ,1 0,1

,1 ,1 0

, ,

, ,

l

M B

S D W

T S f

Area 2

2 ,2 0,2

,2 ,2 0

, ,

, ,

l

M B

S D W

T S f

Figure 3.15. Model of a two area system. Area 1 is much smaller than Area 2.

As Area 2 is very big (infinite bus) it follows that

TM,

2SB2

f0

TM,

1SB1

f0 f2 = constant 2 = 0 (3.28)

and consequently

PT12 = PT21 = PT1 = 2PT

f1dt (3.29)

Without any scheduled generator setpoint changes, i.e. Psetm0,i = 0, thefollowing transfer functions apply for a change in the Area 1 system loadPload,1(s):

f1(s) = s

2PT + (1

Dl,1+

1

SB,1(1 + sTt))s +

TM,1SB,1f0

s2Pload,1(s)

(3.30)

PT12(s) =2PT

sf1(s) (3.31)

PT12(s) =2PT

2PT + (1

Dl,1+

1

SB,1(1 + sTt))s +

TM,1SB,1f0s2

Pload,1(s)

(3.32)


40/119


The response for

Pload,1(s) = Pload,1s (3.33)

is shown if Figure 3.16. The steady state frequency deviation is

f1, = lims0

(s f1(s)) = 0 (3.34)

and the steady state deviation of the tie line power is

PT12, = lims0

(s PT12(s)) = Pload,1 (3.35)

The infinite bus brings the frequency deviation f1 back to zero. Thisis achieved by increasing the tie-line power so the load increase is fully

compensated. While this is beneficial for the system frequency in Area 1,a new, unscheduled and persisting energy exchange has arisen between thetwo areas.

0 1 2 3 4 5 6 7 8 9 1049.85

49.9

49.95

50

Time [s]

Frequency[Hz]

f1

f

2

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

Time [s]

Power

Pload,1

Pm,1

PT12

Figure 3.16. Step response for the system in Figure 3.14 resp. Figure3.15. Only primary control is used and the system is parameterizedaccording to Table 3.1. The upper diagram shows the frequencies f1in Area 1 and f2 in Area 2. The lower diagram shows the step in thesystem load Pload,1, the turbine power Pm,1 and the tie-line powerPT21 = PT12.


41/119



H1 5 sH2 5 s

SB,1 10 GWSB,2 10 TW

f0 50 Hz

Dl,11

200

Hz

MW

Dl,21

200

Hz

TW

W0,1 0MW

Hz

W0,2 0MW

Hz

S11

5000

Hz

MW

S21

5000

Hz

MW

PT 533.33 MWPload,1 1000 MW

Table 3.1. Parameters for time domain simulation of the two-areapower system corresponding to Figure 3.16.


42/119


3.4 Turbine Modelling and Control

This section gives an overview of the modelling of both steam and hydroturbines, control valves and governors. Their characteristics and behaviourare also briefly discussed. The aim here is to give an understanding of thebasic physical mechanisms behind these models that are very commonlyused in simulation packages for the study of power systems dynamics. Inthe dynamic frequency model of the power system derived so far (as depictedin Figure 3.11), the linearized dynamic models of the turbines will enter inthe block Turbine dynamics/control.

3.4.1 Turbine Models

Steam Turbines

Figures 3.17, 3.18, and 3.19 show the most common steam turbines andtheir models. It is outside the scope of these lecture notes to give a de-tailed derivation and motivation of these models, only a brief qualitativediscussion will be provided. In a steam turbine the stored energy of hightemperature and high pressure steam is converted into mechanical (rotat-ing) energy, which then is converted into electrical energy in the generator.The original source of heat can be a furnace fired by fossil fuel (coal, gas, oroil) or biomass, or it can be a nuclear reactor.

The turbine can be either tandem compound or cross compound. In

a tandem compound unit all sections are on the same shaft with a singlegenerator, while a cross compound unit consists of two shafts each connectedto a generator. The cross compound unit is operated as one unit with oneset of controls. Most modern units are of tandem compound type, even ifthe crossover compound units are more efficient and have higher capacity.However, the costs are higher and could seldom be motivated.

The power output from the turbine is controlled through the positionof the control valves, which control the flow of steam to the turbines. Thevalve position is influenced by the output signal of the turbine controller.Following the nomenclature introduced in Figure 3.10, this signal is definedas

Pctrlm =1

Tts(Psetm0 + Psetm Pm) . (3.36)

The delay between the different parts of the steam path is usually modelledby a first order filter as seen in Figures 3.17, 3.18, and 3.19. Certain fractionsof the total power are extracted in the different turbines, and this is modelledby the factors FV HP, FHP, FIP, FLP in the models. Typical values of thetime constant of the delay between the control valves and the high-pressureturbine, TCH, is 0.1 0.4 s. If a re-heater is installed, the time delay islarger, typically TRH = 4 11 s. The time constant of the delay betweenthe intermediate and low pressure turbines, TCO , is in the order of 0.30.6 s.


43/119

3.4. Turbine Modelling and Control 37

Shaft

To Condenser

ctrl

mP mP

Shaft

To Condenser

Valveposition

ctrlmP

mP

ControlValves,Steam

Chest

HP

1

1 CHsT

Nonreheat Steam Turbines

Linear Model

ControlValves,Steam

Chest

HP

Reheater Crossover

IP LP LP

Tandem Compound, Single Reheat Steam Turbines

1

1CH

sT

1

1 RHsT

1

1 COsT

HPF IPF LPF

Linear Model

Valve

Position

Figure 3.17. Steam turbine configurations and approximate linearmodels. Nonreheat and tandem compound, single reheat configura-tions.


44/119


Shaft

To Condenser

Valveposition

Control

Valves,Steam

Chest

VHP

Reheater Crossover

HP LP LP

Tandem Compound, Double Reheat Steam Turbine

1

1 CHsT 1

1

1RH

sT2

1

1 RHsT

HPF

IPF

Linear Model

Reheater

IP

1

1 COsT

HPF

ctrl

mP

mP

VHPF

Figure 3.18. Steam turbine configurations and approximate linearmodels. Tandem compound, double reheat configuration.


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HP, LP ShaftValve

position

IP, LP Shaft

ControlValves,

Steam

Chest

HP

Reheater Crossover

IP

LP LP

Cross Compound, Single Reheat Steam Turbine

LP LP

1

1 CHsT

1

1RH

sT

1

1 COsT

HPF

2LPF

Linear Model

2LPF

IPF

ctrl

mP

1mP

2mP

Figure 3.19. Steam turbine configurations and approximate linearmodels. Cross compound, single reheat configuration.


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Step Response To illustrate the dynamics of a steam turbine, the configu-

ration with tandem compound, single reheat, Figure 3.17, with the followingdata will be studied:

TCH = 0.1 s, TRH = 10 s, TCO = 0.3 s

FHP = 0.3, FIP = 0.4, FLP = 0.3

As TCH TRH und TCO TRH, we can assume TCH = TCO = 0 for anapproximate analysis. Then a simplified block diagram according to Fig-ure 3.20 can be used. For this system the step response is easy to calculate.It is depicted in Figure 3.21.

1

1 10s

0.3 0.7

ctrl

mP

mP

Figure 3.20. Simplified model of tandem compound, single reheatsystem in Figure 3.17.

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Powerfromturbine

FIP + FLP = 0.7

FHP = 0.3

TRH = 10 s

Figure 3.21. Step response of system in Figure 3.20.


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Area A=

Velocity v=

P1

Length of

Penstock L=

h Head=

P2

Effective Area a=

Output Velocity vout=

Figure 3.22. Schematic drawing of hydro turbine with water paths.

Hydro Turbines

Compared with steam turbines, hydro turbines are easier and cheaper tocontrol. Thus, frequency control is primarily done in the hydro power plantsif available. If the amount of hydrogenerated power in a system is notsufficient, the steam turbines have to be included in the frequency control.

The power produced by a generator is determined by the turbine gover-nor and the dynamic properties of the turbine. Thus, to be able to determinethe frequencys dynamic behaviour, models for the turbine as well as for theturbine control are necessary.

Figure 3.22 depicts a hydro turbine with penstock and hydro reservoir

and defines the notation that will be used from now on. Bernoullis equationfor a trajectory between the points P1 and P2 can be written as

P2P1

v

t dr + 1

2(v22 v21) + 2 1 +

P2P1

1

dp = 0 . (3.37)

The following assumptions are usually made:

v1 = 0, since the reservoir is large and the water level does not changeduring the time scale that is of interest here.


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The water velocity is nonzero only in the penstock. The water is incompressible, i.e. does not change with water pressure. The water pressure is the same at P1 and P2, i.e. p1 = p2.Further,

2 1 = gh . (3.38)The above assumptions together with eq. (3.38) make it possible to write(3.37), with vout = v2 and the length of the penstock L, as

L

dv

dt +

1

2 v

2

out gh = 0 . (3.39)The velocity of the water in the penstock is v. The effective opening of thepenstock, determined by the opening of the turbines control valve (guidevanes), is denoted a. If the penstocks area is A,

vout =A

av (3.40)

is valid and eq. (3.39) can be written as

dv

dt

=1

L

gh

1

2LA

a

v2

. (3.41)

The maximum available power at the turbine is

P =1

2av3out =

1

2

A3v3

a2. (3.42)

To get the system into standard form,

x = v ,

u =a

A,

y = P ,

(3.43)

are introduced. (Here, we have used the standard notation, i.e. x for state, ufor control signal, and y for output signal.) The system now can be writtenas

x =gh

L x2 1

2Lu2,

y = Ax3

2u2.

(3.44)


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*/*

u x

x

x u

x2

u2

td

dx

1

2L------

1

s---

A

2-------

gh

L------

y P Ax

3

2u2

--------= =

y-

+

Figure 3.23. Block diagram showing model of hydro turbine.

The system corresponding to eq. (3.44) can be described with the blockdiagram in Figure 3.23.

Eq. (3.44) is nonlinear and a detailed analysis is beyond the scope ofthese lecture notes. To get an idea of the systems properties, the equationsare linearised, and small variations around an operating point are studied.

In steady state, x = 0, and the state is determined by x0

, u0

, and y0

, whichfulfils

x0 = u0

2gh ,

y0 =Ax3

0

2u20

.

(3.45)

Small deviations x, u, and y around the operating point satisfy

x = 2x01

2Lu20

x +2x2

0

2Lu30

u ,

y = 3Ax202u2

0

x 2 Ax302u3

0

u ,

(3.46)

which, using eqs. (3.45), can be written as

x =

2gh

u0Lx +

2gh

u0Lu ,

y =3y0

u0

2ghx 2y0

u0u .

(3.47)


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u

0 5 10 15 20

u1t s( )

y

y0

u0-----u1

2y0

u0------ u1

t s( )

Tw 5 s=

Figure 3.24. The variation of the produced power, y, after a stepchange in the control valve.

The quantity L/

2gh has dimension of time, and from the above equa-tions it is apparent that this is the time it takes the water to flow throughthe penstock if a = A. That time is denoted T:

T = L/

2gh . (3.48)

If eqs. (3.47) are Laplacetransformed, x can be solved from the firstof the equations, leading to

x =L/T

1 + su0Tu , (3.49)

which, when inserted in the lower of eqs. (3.47) gives

y =y0u0

1 2u0T s1 + u0T s

u . (3.50)

The quantity u0T = a0T /A also has dimension of time and is denoted Tw.The time constant Tw is the time it takes for the water to flow through thepenstock when a = a0 or u = u0, i.e. for the operation point where thelinearization is done. Eq. (3.50) can thus be written as

y =y0u0

1 2Tws1 + Tws

u . (3.51)


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It is evident that the transfer function in eq. (3.51) is of nonminimum

phase, i.e. not all poles and zeros are in the left half plane. In this case, onezero is in the right half plane. That is evident from the step response toeq. (3.51), depicted in Figure 3.24.

The system has the peculiar property to give a lower power just afterthe opening of the control valve is increased before the desired increasedpower generation is reached. The physical explanation is the lower pressureappearing after the control valve is opened, so that the water in the penstockcan be accelerated. When the water has been accelerated, the generatedpower is increased as a consequence of the increased flow. That property ofwater turbines places certain demands on the design of the control systemfor the turbines.

3.4.2 Steam Turbine Control Valves

As stated earlier, the control input of the steam turbine acts on a valve whichinfluences the inflow of live steam into the turbine. As the valve cannot bemoved at infinite speed, a further dynamics is introduced into the system.This is usually modelled by a first-order element. Furthermore, constraintsexist on the ramp rate (speed of the valve motion) and the absolute valveposition. Note that, if the block diagram is expressed in quantities, thelatter has to be formulated in quantities as well, taking into account thesteady-state valve position. Figure 3.25 depicts the corresponding block

diagram.Note that this block must be inserted between the turbine controlleroutput Pctrlm and the turbine input when the valve dynamics shall be con-sidered. The output of the valve can be named then Pctrlm and its inputcould be renamed to e.g. Pctrl

m .

1

VT

1

s

*ctrl

mPctrl

mP

upP

downP

,0min min min P P P

,0max max max P P P

Figure 3.25. Model of a control valve.


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3.4.3 Hydro Turbine Governors

The task of a turbine governor is to control the turbine power output suchthat it is equal to a set value which consists of a constant and a frequency-dependent part, the latter of which is of interest here. For steam turbines,this has been discussed in Section 3.1. Because of the particularities of hydroturbines, which have been shown in Section 3.4.1, another type of governoris needed.

A model of a hydro turbine governor is given in Figure 3.26. The controlservo is here represented simply by a time constant Tp. The main servo,i.e. the guide vane, is represented by an integrator with the time constantTG. Typical values for these parameters are given in Table 3.2. Limits foropening and closing speed as well as for the largest and smallest opening ofthe control valve are given.

1

1p

T s

1

s

f u1

GT

openu

closeu minu

maxu

0u

1

R

R

T sT s

u

Figure 3.26. Model of turbine governor for hydro turbine.

Parameter Typical ValuesTR 2.5 7.5 s

TG 0.2 0.4 s

Tp 0.03 0.06 s

0.2 1

0.03 0.06

Table 3.2. Typical values for some parameters of the turbine governorfor hydro power.


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3.5. Dynamic Responses including Turbine Dynamics 47

The controller has two feedback loops, a transient feedback loop and

a static feedback loop. The transient feedback loop has the amplification for high frequencies. Thus, the total feedback after a frequency changeis ( + ). In steady state, the transient feedback is zero, and the ratiobetween the frequency deviation and the change in the control valve is givenby

u = 1

f . (3.52)

Using eq. (3.51), the stationary change of power is obtained as

P = 1

P0u0

f . (3.53)

Thus, the speed droop for generator i, Si, is

Si = iu0iP0i

, (3.54)

and the total speed droop, S, in the system is given by

S1 =i

S1i . (3.55)

The transient feedback is needed since the water turbine is a nonminimumphase system as discussed above. With static feedback only, the controlperformance will be unsatisfactory when closed-loop stability shall be guar-anteed. Increasing the static feedback to a range where a reasonable control

performance could be attained will make the system unstable.The transient feedback loop provides an additional feedback componentduring non-stationary operating conditions. This feedback decays as steadystate is attained. The initial total feedback can be about ten times largerthan the static feedback, i.e. the speed droop is initially lower than its sta-tionary value.

3.5 Dynamic Responses including Turbine Dynamics

In this section, we will simulate the dynamic responses of a power systemconsisting of different generation unit types with primary control equipment.

The goal is to improve the understanding of the impact of the turbine dy-namics on the overall control behaviour of the system.

We consider again the one-area power system as presented in Figure 3.11with the main parameters given according to Table 3.3. As stated before, theturbine dynamics is inserted in the block Turbine dynamics/control. Theeffect of three different turbine dynamics will be studied: a steam turbinewithout reheater, a steam turbine with reheater and a hydro turbine.

In Figure 3.27, the power system response to a load increase disturbanceof Pload = +1000 MW is compared. Note that this is equivalent to theloss of a major generation unit, e.g. a nuclear power plant.


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H 5 sSB 10 GWf0 50 Hz

DL1

200

Hz

MW

W0 0MW

Hz

TCH 0.3 sTRH 8 sTCO 0.5 s

S 0.04 p.u.

TW 1.4 sTG 0.2 s

TR 6.5 s 0.04 p.u. 0.3 p.u.

Table 3.3. Parameters for time domain simulation of the power system.

The upper plot shows the frequency response of the uncontrolled powersystem just as a comparison. It can be seen in the following plots thatthe frequency control performance is highly dependent on the turbine type.While steam turbines without reheating equipment do not allow the fre-quency to decay substantially more than their steady-state deviation, the

time delays caused by the reheater make the response a lot slower. Thehydro turbine allows the most significant frequency decay before the systemis brought back to the steady-state frequency.


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3.5. Dynamic Responses including Turbine Dynamics 49

0 5 10 15 20 25 30

46

48

50

Uncontrolled power system

f[Hz]

0 5 10 15 20 25 3049

49.5

50Primary control on unit without reheating

f[Hz]

0 5 10 15 20 25 3049

49.5

50Primary control on unit with reheating

f[Hz]

0 5 10 15 20 25 3048

49

50Primary control on hydro unit

f[Hz]

0 5 10 15 20 25 300

500

1000

System load step Pload

Pload

[M

W]

Time [s]

Figure 3.27. Dynamic response of the power system without pri-mary control, with primary control on steam turbines without reheater,steam turbines with reheater, and hydro turbines.


56/119


Finally, one more illustration of the effect of the turbine dynamics shall

be given. For a direct comparison of a primary-controlled one-area systemwithout and with turbine dynamics, the exemplary system simulated inFigure 3.12 is now simulated again with a steam turbine model withoutreheater (TCH = 0.3 s). The result is shown in Figure 3.28.

0 1 2 3 4 5 6 7 8 9 1049.75

49.8

49.85

49.9

49.95

50

SystemFrequenc

y[Hz]

0 1 2 3 4 5 6 7 8 9 10

0

100

200

300

400

Time [s]

P

ower[MW]

Pload

Pm

Pload

f

Figure 3.28. Behaviour of the one-area system (including dynamicsof steam turbine without reheater, otherwise same parameterization asin Figure 3.12) after a step increase in load. The upper plot shows thesystem frequency f. The lower plot shows the step function in Pload,the increase in turbine power Pm, and the frequency-dependent loadvariation Pfload.


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4Load Frequency Control

In this chapter the secondary, or load-frequency, control of power systemswill be discussed. Simple models that enable the simulation of the dynamicbehaviour during the action of frequency controllers will also be derived andstudied.

In the previous chapter, the role of the primary frequency control wasdealt with. It was shown that after a disturbance a static frequency errorwill persist unless additional control actions are taken. Furthermore, theprimary frequency control might also change the scheduled interchanges be-tween different areas in an interconnected system. To restore the frequencyand the scheduled power interchanges, additional control actions must betaken. This is done through the Load-Frequency Control (LFC). The LFCcan be done either manually through operator interaction or automatically.In the latter case it is often called Automatic Generation Control (AGC).The characteristics of AGC will be studied in the subsequent sections, both

during steady state and dynamic conditions.

4.1 Static Characteristics of AGC

The overall purpose of the Automatic Generation Control comprises twomain aspects:

Keep the frequency in the interconnected power system close to thenominal value.1

Restore the scheduled interchanges between different areas, e.g. coun-tries, in an interconnected system.

The mechanisms of AGC that enable it to fulfill these requirements will beoutlined in the sequel.

1In many systems, deviations of up to 0.1 Hz from the nominal value (50 or 60 Hz)is deemed as acceptable in steady state. In some systems, North America, even tightertolerance bands are applied, while recently in UK the tolerance band has been relaxedsomewhat.

51


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52 4. Load Frequency Control

G

G G

G

G

GG1TP 2TP

1f 2f

1

1 1Tie line power for Area 1 Sum over all tie lines=j

T T

j

P P

Area 1 Area 2

AGC2AGC1

2

set

AGCP1set

AGCP

Figure 4.1. Two area system with AGC.

Consider a two area system as depicted in Figure 4.1. The two secondaryfrequency controllers, AGC1 and AGC2, will adjust the power referencevalues of the generators participating in the AGC. In an N-area system,

there are N controllers AGCi, one for each area i. A block diagram of sucha controller is given in Figure 4.2. A common way is to implement this as aproportional-integral (PI) controller:

PAGCi = (Cpi +1

sTNi)ei (4.1)

where Cpi = 0.1 . . . 1.0 and TNi = 30 . . . 200 s. The error ei is calledArea Control Error, ACEi for area i.

We will now consider a system with N areas. The ACEs are in this case:

ACEi = ji

(PjT i PjT0i) + Bi(f f0) i = 1, 2, . . . ,N . (4.2)

Defining now

PTi =ji

(PjT i PjT0i) , (4.3)

the ACE can be written as

ACEi = PT i + Bif i = 1, 2, . . . , N . (4.4)

The set i consist of all areas connected to area i for which the tie linepowers should be controlled to the set value PjT0i. The constants Bi are


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4.1. Static Characteristics of AGC 53

TiP

0T iP

0f

f

AGCiPie

Bi

AGCi

f

Figure 4.2. Control structure for AGC (ei = Error = ACEi =Area Control Error for area i)

called frequency bias factors [MW/Hz]. It is assumed that the frequencyreferences are the same in all areas, i.e. f0i = f0 for all i, and f is also the

same in steady state for all areas. The goal is to bring all ACEi 0.The variables are thus PTi (N variables) and f, i.e. in total N + 1 vari-ables. Since we have N equations (ACEi = 0), we need one more equation.As a fifth equation we have the power balance:

i

PTi = 0 , (4.5)

and consequently a solution can be achieved.

In steady state, f is identical for all areas, and we assume that thefrequency is controlled back to the reference value, i.e. f0 = f. If the sumof the reference values of the tie line powers PjT0i is 0, then the system will

settle down to an operating point where PjT0i = PjT i for all tie line powers.The time constants of the AGC is chosen such that it reacts much slowerthan the primary frequency control.

Selection of Frequency Bias Factors

Consider the two area system in Figure 4.1. The load is now increased withPload in area 2. If the tie line power should be kept the same, the generationmust be increased in area 2 with Pload after the AGC has reacted. Beforethe AGC has reacted we have a frequency deviation of f in both areas,


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which means that for area 1

f = S1PT1 (4.6)and for area 2

f = S2(Pload PT1) (4.7)and PT1 = PT2. The two ACEs can now be written as

ACE1 = PT1 + B1f = PT1 + B1(S1PT1) = PT1(1 B1S1) (4.8)and

ACE2 = PT2 + B2f = PT2 + B2(S2(Pload PT1)) =PT2(1 B2S2) B2S2Pload (4.9)

In this case it is desirable that the AGC controller in area 1 does not react.If we set B1 = 1/S1 we see from eq. (4.8) that ACE1 = 0. This is calledNon Interactive Control. IfB2 = 1/S2 is chosen the ACE in area 2 becomes

ACE2 = Pload (4.10)This means that only controller 2 reacts and the load increase Pload iscompensated for in area 2 by the PI control law as stated in eq. (4.1).

However, as long as the controller in eq. (4.1) has an integrating part,all positive values of Bi will guarantee that all ACEi 0. The choice ac-cording to Non Interactive Control has been found to give the best dynamicperformance through a number of investigations. In a multi-area case this

corresponds to selecting Bi = 1/Si for all areas.

4.2 Dynamic Characteristics of AGC

As in the previous chapter on primary frequency control, we will now developa dynamic model of the power system where the newly introduced AGC isincluded. As the way of extracting input-output transfer functions fromblock diagrams has already been presented in the last chapter, we omit theanalytical derivation here for shortness. The interested reader is welcometo calculate the transfer function between the load disturbance and thefrequency as an exercise.

4.2.1 One-area system

In a single-area power system, there is obviously no tie-line power to becontrolled. In this case, the AGC only fulfills the purpose of restoring thenominal system frequency. Figure 4.3 shows the known block diagram whichhas been extended by the proportional-integral control law of the AGC.Figure 4.4 shows a plot of the system frequency which is brought back tothe nominal value. The parameters of the system are as presented in Table3.3 (turbine dynamics neglected), and B = 1/S, Cp = 0.17, TN = 120 s.


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4.2. Dynamic Characteristics of AGC 55

loadP

f

eP

loadP

0

0

2Ws

f

Rotating mass loads


System inertia

System load change1

lD

0

(2 )B

f

HS s

mP

1

1tT s

set

mP

0

set

mP

1

S

Turbine

dynamics/

control

Primary

control

1)( p

N

Cs T

ACEAGCPschedP

AGC

(one area

system)B

frequency

bias

Figure 4.3. Dynamic model of one-area system with AGC.

0 100 200 300 400 500 600 700 800 90049.7

49.75

49.8

49.85

49.9

49.95

50

50.05

50.1

Time [s]

f[Hz]

with ACG

without AGC

Figure 4.4. Dynamic response of the one-area system equipped with

primary control and AGC compared with the same system withoutAGC.


62/119


4.2.2 Two-area system unequal sizes disturbance response

Here we consider the same two-area power system as in section 3.3.4 whereArea 1 is assumed to be much smaller than Area 2. The corresponding blockdiagram including the AGC is shown in Figure 4.6. Because of the secondarycontrol (AGC) one obtains the step response in Figure 4.5. The load increaseof Pload,1 = 1000 MW in Area 1 is in this case fully compensated by thegenerators in Area 1.

0 100 200 300 400 500 60049.8

49.85

49.9

49.95

50

50.05

Systemfreq

uency[Hz]

f1

f2

0 100 200 300 400 500 600200

0

200

400

600

800

1000

Time [s]

P

[MW]

PAGC,1

PAGC,2

PT21

Figure 4.5. Step response for the system in Figure 4.6 with AGC. Theupper diagram shows the system frequencies of Area 1 and Area 2. Thelower diagram shows the control a

48461536-DynCtrl-FS2010-notes-part-1

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