Modelling of Load Devices and Studying Load/System Characteristics

Modelling of Load Devicesand

Studying Load/System Characteristics

Kerstin Lindénand

Inger Segerqvist

Technical Report No. 131LDepartment of Electrical Power Systems

School of Electrical and Computer Engineering1992

School of Electrical and Computer Engineering

CHALMERS UNIVERSITY OF TECHNOLOGY

Göteborg, Sweden

Technical Report No. 131L

(Revised edition)

Modelling of Load Devicesand

Studying Load/System Characteristics

by

Kerstin Lindén and Inger Segerqvist

Submitted to the School of Electrical and Computer Engineering,

Chalmers University of Technology,

in partial fulfilment of the requirements for the degree of

Licentiate of Engineering

Department of Electrical Power Systems

Göteborg, Augusti 1992

GOTEBORG

CH

AL

ME

RSTEKNISKAHOG

SKO

LA

CHALMERS TEKNISKA HÖGSKOLA

Institutionen för Elkraftsystem

S–412 96 GÖTEBORG, Sweden

ISBN 91–7197–038–X

Chalmers Bibliotek

ReproService

Göteborg, Oktober 1993

Abstract

This thesis deals with deriving load models from laboratory measurements andstudying the impact of tap-changers on both load and system characteristics.

The laboratory measurements were carried out on various types of lighting,thermostat regulated radiators and refrigerators. The results show that

lamps

have

no pronounced dynamic voltage dependence

and that a large group of the testedlamps have an almost

quadratic static voltage dependence

in active power. Bothfluorescent and mercury lamps have also a strong voltage dependence in reactivepower.

The bimetallic regulated radiators

act as a constant impedance load during a timeperiod of tens of seconds, but have a pronounced dynamic behaviour in a time rangeof minutes. Unexpectedly, the mean power was changed in an opposite direction tothe voltage, i.e.

when the voltage step was negative the mean power increased

andvice versa, and the original mean power as it was prior to the disturbance was notregained. This effect is explained by two resistors in the bimetallic switch.

The electronically regulated radiators

also act as a constant impedance load duringa time period of tens of seconds. The dynamic behaviour after a voltage step, in atime range of minutes, is a

damped oscillation in mean power

, where the prior meanpower is regained.

Refrigerators

show

no dynamic voltage dependence

and active power has a staticvoltage dependence between constant current load and constant impedance load.Refrigerators have a poor power factor, are uncompensated, and the reactive powerhas a rather high voltage dependency.

The studies on the impact of tap-changers on load and system characteristics areperformed using load characteristics and the well-known up-curves. The studies areintended to elucidate the processes described (an increase in load and a decrease infeeding voltage) in an illustrative way.

Key words

Load devices, load model, power system, up-curve, load characteristics, laboratorymeasurements.

Acknowledgement

We would like to thank our supervisor, Professor Bertil Stenborg, for his guidanceand encouragement throughout this project.

Special thanks to Kenneth Walve, Svenska Kraftnät, for the idea of this project andencouraging support, and to Bert Lanne, Chalmers University of Technology, forsharing his knowledge of measuring.

Everyone at the Department of Electrical Power System and the Department ofHigh Voltage are warmly acknowledged for creating a most friendly atmosphere.Special thanks to Jan-Olov Lantto for his help when producing this thesis, and toJan Andersson for his help with the field measurements.

We would like to express our gratitude to Vattenfall, Transmission, and to SvenskaKraftnät who took over the responsibility after the reorganization of Vattenfall, forthe financial support. Special thanks to manager Dag Holmberg and Per-OlofLindström for their interest in this project.

Finally, we thank El och Trafikteknik AB and Göteborg Energi AB for letting ususe their network in the field measurements and for furnishing us with street lamps,and Osram and Philips who provided us with lamps as test objects.

Introduction

One of the fields of research presenting greatest international interest is the studyand analysis of voltage condition in power systems. The research is aimed towardsproblems concerning voltage stability and “voltage collapse”, as well as problemsof transient stability. It is necessary to have sufficiently good models of how theload varies with voltage and frequency in order to carry out an analysis of theseproblems.

The aim of this thesis is to increase the knowledge in load modelling - to deriveaccurate static load models and to estimate time constants for different loadcomponents. Furthermore the impact of on-load tap changers is studied.

Chapter 1 gives a presentation of the load modelling definitions recommended byIEEE task force on Load Representation for Dynamic Performance and a discussionon some commonly used static and dynamic load models.

Chapter 2 takes a look at some statistics on load compositions and the occurrenceof the load devices tested in the laboratory at the Department of Electrical PowerSystems at Chalmers University of Technology in Göteborg.

Chapter 3 presents laboratory measurements on different types of lighting, radiatorsand refrigerators. Both static and dynamic measurements are performed in order toderive static load models and to identify the time-constants for the load components.

Chapter 4 presents results from a field measurement on street lighting, and theresults from the laboratory measurements are compared to the results from a fieldmeasurement on a composite load.

In Chapter 5 the impact of on-load tap changers on both system and loadcharacteristics are discussed. In Section 5.5 the studies are based on the ideas ofProfessor B. Stenborg.

Symbols and graphics

The symbols used for quantities and units coincide with the IEC recommendations,except for the decimal sign which in this thesis is represented by a

dot

, and voltagewhich in this thesis is represented by U and u.

Irregularities in the curves in Chapter 5 are due to the insufficient resolution of theplotting routines of the computer graphics application.

Contents

Abstract

i

Acknowledgement

iii

Introduction

v

Contents

vii

1 An introduction to load modelling

1

1.1 Introduction1

1.2 Definitions1

1.3 Load models2

1.3.1 Static load models3

1.3.2 Dynamic load models5

1.3.3 How to derive parameters7

1.4 References9

2 Load composition

11

2.1 Introduction11

2.2 Identifying a composite load11

2.3 Load components13

2.3.1 Lighting13

2.3.2 Electric heating13

2.3.3 Refrigerator/Freezer14

2.4 References15

3 Laboratory measurements on various loads

17

3.1 Introduction17

3.2 Data acquisition system17

3.3 The static measurements19

3.4 The dynamic measurements19

3.5 Accuracy analysis19

3.6 Laboratory measurements on lamps20

3.6.1 Test objects20

3.6.2 Results from static measurements21

3.6.3 Results from dynamic measurements22

3.6.4 Accuracy analysis243.6.4.1 Harmonics generated by discharge lamps27

3.6.4.2 The inaccuracy due to harmonics30

3.6.4.3 The definition of reactive power under non-sinusoidal conditions31

3.6.5 The static load models32

3.6.6 The dynamic behaviour34

3.6.7 Discussion35

3.7 Laboratory measurements on electric heating devices36


3.7.2 The climate box38

3.7.3 Results39

3.7.4 Load model41

3.7.5 Discussion41

3.8 Laboratory measurements on refrigerators42


3.8.2 Results from static measurements43

3.8.3 Results from dynamic measurements44

3.8.4 Accuracy analysis44

3.8.5 The static load models45

3.8.6 Discussion47

3.9 References48

4 Field measurements

49

4.1 Introduction49

4.2 Field measurements on street-lamp networks49

4.2.1 Test net49

4.2.2 Data acquisition system50

4.2.3 Dynamic measurements51

4.2.4 Accuracy51

4.2.5 Results52

4.2.6 Discussion55

4.3 Field measurements on composite load56

4.4 References57

5 Voltage stability with regard to system and load characteristics

59

5.1 Introduction59

5.2 System characteristics59

5.3 Load characteristics63

5.4 Tap changer influence on load characteristics64

5.4.1 Increase in load64

5.4.2 Decrease in feeding voltage67

5.4.3 Discussion695.4.3.1 Increase in load69

5.4.3.2 Decrease in feeding voltage71

5.5 Tap changer influence on system characteristic72

5.5.1 Increase in load74

5.5.2 Decrease in feeding voltage75

5.5.3 Discussion76

5.6 Reflections on some simplification77

5.6.1 Resistance77

5.6.2 Relation between the active and the reactive power.80

5.7 References81

6 Future work

83

Bibliography

85

Appendix

A Street Lighting

87

B Inaccuracy due to measurement devices and measurementarrangement

91

C Dynamic voltage tests

95

D Accuracy analysis

103

E Wave forms

111

F Derived static load models

115

G Evaluation of the equations for up-curves and qu-curves

127

H Evaluation of the equations for the impact of tap changer onsystem characteristics

133

I Exponential load model

137

xi

Ryggtext:

K Lind n & I Segerqvist Modelling of Load Devices and Studying

Chapter 1: An introduction to load modelling

1

Chapter 1

An introduction to load modelling

1.1 Introduction

This chapter deals with basic load modelling definitions recommended by IEEETask force on Load Representation for Dynamic Performance [1] and a discussionon some commonly used static and dynamic load models based on the references ofthis chapter and the bibliography listed at the end of this thesis. The advantages anddisadvantages of component based or field measurement based methods arediscussed, and some parameters for different load objects are also presented.

1.2 Definitions

Following definitions described in ref. [1] are used in this thesis.

Load

The term “load” is a wide conception:

1. A device, connected to a power system, that consumespower. To clarify, the term “load device” may be usedfor this “type”.

2. The total power (active and/or reactive) consumed by alldevices connected to a power system. To clarify, theterm “system load” may be used for this “type”.

3. A portion of the system that is not explicitly representedin a system model, but rather is treated as if it were asingle power-consuming device connected to a bus in thesystem model. This includes tap changers, capacitors,distribution network etc. To clarify, the term “bus load”may be used for this “type”.


2

4. The power output of a generator or generating plant. Toclarify, the term “generator or plant load” may be usedfor this “type”.

In describing the composition of the load, this thesis also uses the recommendationsfrom ref. [1].

Load component

A load component is the aggregate equivalent of all devicesof a specific or similar type, e.g. fluorescent lighting.

Load class

A load class is a category of load, such as residential,commercial or industrial.

Load composition

A load composition is the fractional composition of the loadby load components.

Load class mix

A load class mix is the fractional composition of the busload by load classes.

Load characteristic

Load characteristic is determined by a set of parameters thatcharacterise the behaviour of a specified load, such as thevariation of the active power P with voltage U. This termmay be applied to a specific load device, a load component,a load class or the total bus load.

1.3 Load models

A load model in this matter is a mathematical representation of the relationshipbetween power and voltage, where the power is either active or reactive and theoutput from the model. The voltage (magnitude and/or frequency) is the input to themodel. The load model could be a static or dynamic load model or a combination ofboth. Load models are used for analysing power system stability problems, such assteady state stability, transient stability, long term stability and voltage control.

It isnot said that the same load model is appropriate for different stability analysis. Inorder to obtain a model which is as simple as possible, it is important to choose aload model structure which is appropriate for the studied problem

, for example iftap changers and time dependence should be included. A rough guideline forchoosing a load model structure is presented in a CIGRÉ paper [2].


3

1.3.1 Static load models

A static load model expresses the active and reactive powers as a function of thevoltage (magnitude and/or frequency). The load model could be a stationary orquasi-stationary representation of the load. The following models are commonlyused

Constant power

A load model, where the active and reactive powers are independent of variationsin the voltage magnitude:

Constant current

A load model, where the active and reactive powers vary directly with the voltagemagnitude:

Constant impedance

A nonlinear load model, where the active and reactive powers vary with the squareof the voltage magnitude:

Polynomial

A nonlinear load model, where the active and reactive power variations to voltagemagnitude are usually a combination of the three mentioned above:

where a

0

, a

1

, a

2

and b

0

, b

1

, b

2

are constants and parameters of the load models. Thesum of the parameters equals one, i.e.:

a

0

+ a

1

+ a

2

= 1 b

0

+ b

1

+ b

2

= 1

as the parameters indicate how nominal power is divided into constant power,constant current and constant impedance loads.

PP0------

UU0-------

01= = Q

Q0-------

UU0-------

01= =

PP0------ U

U0-------= Q

Q0------- U

U0-------=

PP0------

UU0-------

2= Q

Q0-------

UU0-------

2=

PP0------ a0 a1

UU0------- a2

UU0-------

2+ += Q

Q0------- b0 b1

UU0------- b2

UU0-------

2+ +=


4

Exponential

A nonlinear load model which represents the power relationship to voltage as anexponential equation:

where np and nq are constants and the parameters of the load models. Note thatconstant power, constant current and constant impedance are special cases of theexponential model.

Slope values

A load model, where the power-voltage dependence is linearised at nominalvoltage:

Frequency dependent

A static load model which includes frequency dependence. This is usuallyrepresented by multiplying either a polynomial or exponential load model by afactor including the frequency deviation and the frequency sensitivity parameter.The factor is usually in the following form:

[1 + af (f - f0)]

where f is the frequency of the bus voltage, f0 is the rated frequency, and af is thefrequency sensitivity parameter of the model. The total static model results forexample in:

PP0------

UU0-------

np

= QQ0-------

UU0-------

nq

=

∆P∆U-------- ∆Q

∆U--------

PP0------

UU0-------

np

1 af f f0–( )+[ ]=


5

1.3.2 Dynamic load models

The interest for dynamic load models has increased during the last decade. Thedynamic load model describes the time dependence as well as the voltagedependence of the load. Listed below are the most commonly used dynamic loadmodels:

Induction motor model

A considerable part of the load consists of machines, especially induction motors.Hence a model for an induction motor is commonly used. The model presented hereis the one often used [2].

Figure 1.1

The inclusion of mechanical dynamics in the model transforms it from static todynamic. This is the most usual linearized form of the equation:

where

∆w is the per unit speed change

∆Ta = Te - Tm, the accelerating torque

Te the electrical torque

Tm the mechanical torque

D the mechanical damping

H the motor and drive inertia

R1

R′2s

jX1 jX2

jXm

∆w1

2Hs---------- ∆Ta D∆w–( )=


6

s = d/dt

where

is the motor torque/speed relationship

is the load torque/speed relationship

State space model

The standard equations in state space form are

where x is the state vector, u the input vector and y the output vector. The modelmay be of order n, but minimum order should be n = 2, to account for the dynamicsproduced by the induction motor swing equation. The state vector x has no physicalmeaning. The output vector y may be chosen as the active and reactive power PLand QL, or as the active and reactive component of load current IR and IIrespectively. The model could for instance look like that in ref. [3]:

DδTe

δw---------

δTm

δw-----------+=

δTe

δw---------

δTm

δw-----------

x Ax Bu+=

y Cx Du+=

x1

x2

0 1

a1– a2–

x1

x2

1 0

0 1

UR

UI

+=

IR

II

c1 c2

c3 c4

x1

x2

d1 d2

d3 d4

UR

UI

+=


7

Transfer function model

The power is represented as a transfer function of the general form

(describes the load-voltage dependence) (1.1)

(describes the load-frequency dependence)(1.2)

The total change in active power is given as

∆PT(t) = ∆PF(t) + ∆PU(t) (1.3)

where ∆PF(t) and ∆PU(t) are obtained by solving Equation (1.1) and Equation (1.2)in the time domain. The reactive power is equally represented. The model and thederivation of parameters are described in ref. [4] and ref. [5].

1.3.3 How to derive parameters

Once a model structure is chosen, see Section 1.3, there remains the difficult task ofderiving the model and its parameters. The characteristic of a bus load depends onthe load composition, which means that the aggregated load characteristics for thebus load must be found. These load parameters can be derived with a so-calledcomponent based method or by a field measurement based method. The firstmethod is based on the knowledge of the load class mix, the composition of eachclass and the characteristics of each load component. Where the characteristics ofload components are not known, they may be derived either theoretically or bylaboratory measurements. The component based approach has been developed in acomputer programme, The Load Model Synthesis, LOADSYN by EPRI [6, 7]. Thesecond method is based on direct measurement at a bus, during system disturbancesor planned system disturbances, where voltage, frequency, active power andreactive power are measured and then a method, such as the Least Square Methodis used to derive parameters to the aggregated load model.

Here is a brief comparison of the component based method with the fieldmeasurement based method:

Component based method

+ Represents any arbitrary load composition, i.e. adaptable to different busloads and conditions (time of day, weather, season etc.)

– Relies heavily on the availability of a large number of data

∆PU s( )∆U s( )------------------ HU s( )=

∆PF s( )∆F s( )

----------------- HF s( )=


8

Field measurement based method

+ Straightforward: simply measure and derive a model

– Unless the load composition is analysed in some detail and unless buseshaving loads of fairly different compositions are measured there will beno understanding of the results so that they can be extrapolated todifferent conditions

– The model depends on the disturbance, that is the input, applied to theload

– Spontaneous load variations are included in the load model, especiallyduring long term measurements

Presented below are some parameters for different load devices using the staticexponential load model [8]:

np nq

Incandescent lamp 1.6 0

Fluorescent lamp 1.2 3.0

Heating 2.0 0

Induction motor, half load 0.2 1.5

Induction motor, full load 0.1 2.8


9

1.4 References

[1] IEEE Task force on Load Representation for Dynamic Performance, ‘LoadRepresentation for Dynamic Performance Analysis’ IEEE/PES 1992Winter meeting, New York, January 26 - 30, 1992 (92 WM 126-3 PWRS)

[2] F. McDyer, F. Byrne, R. W. McGee, G. Rodgers, M. Hayashi, G. Testud,N. Roelefs, ‘Load modelling and dynamics’, Electra n 130 May 1990 p.122 - 141

[3] F. John Meyer, Kwang Y. Lee, ‘Improved Dynamic Load Model for PowerSystem Stability Studies’, IEEE Transactions on Power Apparatus andSystems, Vol. PAS-101, No 9, September 1982, 3303 - 3309

[4] S.A.Y Sabir, D.C. Lee, ‘Dynamic load models derived from data aquiredduring system transients’, IEEE Transactions on Power Apparatus andSystems, Vol. PAS-101, No 9 September 1982, 3365 - 3372

[5] D. Karlsson, T. Pehrsson, ‘A dynamic power system load model andmethods for load model parameter estimation’, Technical report No 22L1985, Dep. of Electrical Power Systems, Chalmers University ofTechnology, Sweden.

[6] W. W. Price, K. A. Wirgau, A. Murdoch, J. V. Mitsche, E. Vaahedi, M. A.El-Kady, ‘Load modeling for power flow and transient stability computerstudies’, IEEE Transactions on Power Systems, Vol.3, No. 1, February1988, 180 - 187

[7] E. Vaahedi, H. M. Zein El-Din, W. W. Price, ‘Dynamic load modeling inlarge scale stability studies’, IEEE Transactions on Power Systems, Vol.3,No. 3, August 1988, 1039 - 1045

[8] G. J. Berg, ‘Power-system load representation’ Proceedings IEE Vol. 120,No 3, March 1973, 344 - 348


10

Chapter 2: Load composition

11

Chapter 2

Load composition

2.1 Introduction

In Chapter 1, “An introduction to load modelling”, two ways of deriving a loadmodel and its parameters are described: the component based method and the fieldmeasurement based method. Irrespective of the method used, it is essential to haveas good a knowledge as possible of the load composition, in order either to use theknowledge when composing the model or to understand the results ofmeasurements taken to derive the model.

2.2 Identifying a composite load

There are of course differences in load composition depending on the area studied.In a small area, and correspondingly a low voltage level, the load composition iseasier to identify but the composition is also more individual than when the area islarger and on a higher voltage level, where it is more appropriate to assume ageneral composition.

When identifying a composite load it is usual to divide the load into load classes orload components and distribute the power need to the groups. Commonly used loadclasses in Sweden are industrial-, commercial- and residential loads. Loadcomponents are heating, lighting, motor load and fridge/freezer.

The easiest way of estimating the division of power between the various loadclasses or load components, is to base it on the division of energy consumption. Thisinformation can be obtained from the utilities or from SCB, Statistics Sweden,which collects and compiles data from all energy utilities in Sweden. Table 2.1shows the division of energy consumption on Gotland [1], in Stockholm [2] and inthe whole of Sweden - based on data obtained from SCB [3].


12

Table 2.1 Division of electrical energy consumption for different areas. In the study on

Gotland [1] electrical heating forms a special load class.

By comparing this average consumption to the consumption in heavy and light load

conditions for the load classes it is possible to obtain a division of power for a

specific situation, e.g. in the way described in the Gotland study [1]. With statistics

concerning the composition of load components for each load class, the total power

divided among load components may be obtained. Several studies have been made

of the energy consumption in households, offices, industries etc. during the last

years, [4, 5, 6], and it is possible to use the results from these studies to calculate the

power division. Table 2.2 shows the division of power consumption for load

components on Gotland.

Table 2.2 The electrical power consumption on Gotland [1] in heavy and light load

conditions divided among load components.

In the Stockholm-study [2] climate, lighting, process and other were used as load

components. Electrical heating is included in the climate group, motor and

fridge/freezer in the process-group. Table 2.3 shows the division of energy in

Stockholm.

Table 2.3 Electrical energy consumption in Stockholm divided among load components.

Area electricalheating [%]

residential [%]

commercial [%]

industrial [%]

Gotland 30 15 15 40

Stockholm 38 50 12

Sweden1

1. In this area the electrical energy consumption for railroad and electricity-, gas-, heatingand waterworks are not included.

25 23 42

Area electricalheating [%]

lighting [%]

motor [%]

fridge/freezer [%]

High-load 40 10 40 10

Light-load 20 10 60 10

Area climate [%] lighting [%]

process [%]

other [%]

Stockholm 40 10 40 10


13

2.3 Load components

Here follow some statistics concerning the load components that have been testedin laboratory measurements presented in Chapter 3.

2.3.1 Lighting

Lighting adds up to about ten per cent of total active power consumption. It consistsmainly of incandescent lamps and discharge lamps. But now also low-energylamps, introduced in the middle of the eighties, render an increasing interest.According to manufacturers [7], today nearly 600 000 are sold annually with anincrease of about 25 to 30 per cent each year. Sales figures for low-energy lampsare still well below those for incandescent lamps, which sells around 7 million peryear in Sweden.

Discharge lamps are mainly found in street-lighting. A preliminary inventory ofGöteborg´s street-lighting, by Göteborgs Energi AB, shows how different kinds ofdischarge lamps and other lamps are distributed on a total load of 12.7 MW, Table2.4. In Appendix D the size and power consumption of lamps in the Göteborg street-lamp network is presented.

Table 2.4 The table shows how different kinds of discharge lamps and other lamps are

distributed on the total load in the street-lamp network in Göteborg.

2.3.2 Electric heating

Households, one- or two-dwelling houses with electrical heating (both direct andindirect electrical heating) account for about 15 per cent of the total electric energyconsumption in Sweden. About 40 per cent of these households have directelectrical heating as the only means of heating [8]. In these households about 60 percent of the total electric energy is consumed by the electric heating equipment [5].Most of the annual energy consumption for electrical heating is consumed duringOctober to April, with only about 11% in the period May to September.

Mercury lamp1 [%]

1. At present, when electric fittings in new areas are installed, high pressuresodium lamps are used instead of mercury lamps.

73

High pressure sodium lamp1 [%] 16

Low pressure sodium lamp [%] 8.5

Incandescent-, mixed colour-, metal halogen-and fluorescent lamps [%]

2.5


14

About 50% of the households with direct electrical heating were built during theseventies. For some years in this period, approximately 1 000 000 radiators weresold annually in Sweden, most of them with a bimetallic thermostat, which was themost common thermostat on market. Today about 40 000 radiators are sold eachyear. A new type of thermostat, an electronic temperature regulator, has beendeveloped but still most of the radiators that are sold have the bimetallic thermostat,which will continue to be the most common thermostat for years ahead [9].

2.3.3 Refrigerator/Freezer

In an electrically heated household about 20 per cent of the total electric energyconsumption is accounted for by refrigerator and freezer [5], with refrigerator andfreezer consuming equal amounts. Unlike heating equipment, the energyconsumption for refrigerators and freezers is equally distributed throughout theyear.


15

2.4 References

[1] T. Adielsson, ‘Analys av elleveranssäkerheten för Gotland Modellering avden elektriska belastningen på Gotland’, Vattenfall Report ÖS-015 (Oct1989), in Swedish

[2] Långtidsprognos El 1990 - 2010, Stockholm Energi (1990), in Swedish

[3] ‘Electric energy supply and district heating 1989’, Statistics Sweden,Statistical Report E11:9101, 1991

[4] Belastningsberäkning med typkurvor, Svenska Elverksföreningen (1991),in Swedish

[5] Mätning av hushållens elkonsumtion, uppdelad på de viktigasteelapparaterna, Rapport 2, Rationell elanvändning, FUD Vattenfall (1987),in Swedish

[6] Industrins energianvändning, Energikonsult, Ångpanneföreningen, at therequest of Dept. SM Vattenfall (1984), in Swedish

[7] Ny Teknik - TEKNISK TIDSKRIFT. 1990:14, page 5, in Swedish

[8] ‘Energy statistics for one- or two-dwelling houses in 1990’, StatisticsSweden, Statistical Report E16:9102, 1991

[9] C. Norberg, ‘Direktverkande elradiatorers reglering och konstruktion’,Vattenfall Utveckling AB, Rapport F-90:5 (1990), in Swedish


16

Chapter 3: Laboratory measurements on various loads

17

Chapter 3

Laboratory measurements on various loads

3.1 Introduction

In order to derive the voltage dependence of some individual load objects,measurements were performed in the laboratory of The Department of ElectricalPower Systems at Chalmers University of Technology. In the laboratorymeasurements both static and dynamic voltage tests were carried out on differenttypes of lamps, electrical radiators, and refrigerators. The static voltage tests wereperformed in order to derive and compare three different static load models - theexponential, the linear and the polynomial load model. The dynamic voltage testswere performed on all test objects in order to ascertain if the static models wereapplicable for these objects in the time range of seconds to minutes (not accountingfor fast transient behaviour), or if they had a longer dynamic performance, so thatthe time constants could be identified.

3.2 Data acquisition system

In both the static and dynamic tests a data acquisition system, controlled by acomputer was used. Measurement test results were collected from two multimeterswhich measure Urms and Irms respectively and a wattmeter which measures activepower P. The measurement results were saved on a mass storage device. The dataacquisition system was connected as per Figure 3.1 and the measurementarrangement as in Figure 3.2.


18

Figure 3.1 Data acquisition system.

Figure 3.2 Measurement arrangement.

Multimeter

Voltage

Multimeter

Current

Wattmeter

Active power

Computer

Printer

G

W

VTestobject

A


19

3.3 The static measurements

In the static measurements the voltage was varied between 70% and 110% ofnominal voltage (230 V). In order to avoid systematic errors in the performance ofthe measurement (i.e. in the way it is done), each static measurement comprisedthree test series, the first test series was from nominal voltage down to 150 V, thesecond from 155 V up to 255 V, and the third from 255 V down to 235 V. The testseries were performed in the following manner: Every tenth volt, five datums ofvoltage, current and active power were recorded and the calculated average valueswere saved on the mass storage device. Reactive power Q was calculated from thevoltage U, current I and active power P for each average value, using the equations:

S = U · I (3.1)

(3.2)

3.4 The dynamic measurements

In the dynamic measurements voltage changes were applied to the load, and thevoltage, current, active power and time were measured approximately every secondfor about quarter of an hour or half an hour depending on the test object. The voltagechanges were created by quickly turning an adjustable transformer which resultedin voltage steps of around ±10% or ±20% with a rise time of approximately 0.1–0.2seconds. The reactive power was calculated in the same manner as for the staticmeasurements.

3.5 Accuracy analysis

The error in the measured values and in the calculated values of the reactive poweris dependent on the magnitude of measured values. Since it is the curve form, i.e.the relationship between the measured values, which is of interest in the quasi-stationary measurement, and since the measured values in the quasi-stationarymeasurements vary over a wide range, it is of interest to see how the bias(systematic error) of the recorded values varies over the range. The accuracyanalysis in Appendix B was carried out on the measurement devices and on themeasurement arrangement, and shows that the bias due to the measurementarrangement is negligible. The maximum limit of the absolute error, due to the biasin the measurement devices, is presented for the various loads in each section of thelaboratory measurements.

Q S2

P2

–=


20

3.6 Laboratory measurements on lamps

This section deals with laboratory measurements on the three different main typesof lighting: incandescent lamps, low-energy lamps and discharge lamps (wheredischarge lamp is a generic term for fluorescent lamps, mercury lamps, highpressure sodium lamps and low pressure sodium lamps). Results from static anddynamic voltage tests are presented, including derived static load models, observedswitch-off voltage and time-constants for each type of lamp. In the diagrams usingper-unit values presented in this section, P0, Q0 and U0 are used as base-values,where P0 and Q0 are measured/calculated powers at nominal voltage U0 (~230 V).The load models should be applicable in the time range of seconds to minutes, notaccounting for fast transient behaviour.

3.6.1 Test objects

Six types of lamps were tested:

1. Incandescent lamp

2. Fluorescent lamp

3. Mercury vapour discharge lamp (Mercury lamp)

4. High pressure sodium vapour lamp (High pressure sodium lamp)

5. Low pressure sodium vapour lamp (Low pressure sodium lamp)

6. Low-energy lamp (which is an electronically controlled fluorescent lamp)

New lamps, as well as old, with different power ratings were tested. With dischargelamps (2–5), it is not the lamp as a single object that is of interest, but the wholecircuit consisting of lamp, reactor, capacitor and, for sodium lamps and fluorescentlamps, the igniter too. Therefore, in the discharge lamp tests the test objects werecircuits, all connected according to the manufacturers. It should be noted that alldischarge lamps have a compensating capacitor connected in parallel, except thelow pressure sodium lamp, which has a capacitor in series with the lamp. See thecircuits (test objects) in Figure 3.3 for mercury lamp and low pressure sodium lamp.


21

Figure 3.3 Two test objects: Mercury lamp with reactor and capacitor, and low pressure

sodium lamp with reactor, capacitor and igniter.

3.6.2 Results from static measurements

Figure 3.4 shows the quasi-stationary voltage dependence for the active powerconsumed by an incandescent lamp. The rest of the results can be seen in AppendixF where the recorded values are plotted in the same diagram as the static modelderived later on in Section 3.6.5.

Figure 3.4 The quasi-stationary voltage dependence for the active power of an

incandescent lamp.

Discharge lamps are very sensitive to sudden voltage changes, especially when thevoltage is low, so in order to obtain the switch-off voltage the adjustable transformer

Lamp

ReactorL

N

Capacitor

Mercury lamp

LampReactor

L

NCapacitor

Low pressure sodium lamp

Igniter


22

was turned slowly. Table 3.1 shows both the recorded active power, just before thelamp is switched off, and the switch-off voltages. After the switch-off the circuit ofthe discharge lamp will act as a totally capacitive load due to the compensatingcapacitor (provided the compensating capacitor is in parallel).

Table 3.1 The switch-off voltage and active power just before the switch-off.

3.6.3 Results from dynamic measurements

The following figures (3.5 – 3.7) show consumed active and reactive power as afunction of time when a voltage step of magnitude –10% is applied to anincandescent lamp, a mercury lamp and a high pressure sodium lamp. Note that anincandescent lamp is totally resistive. The results for the other types of lamp can beseen in Appendix C.

Figure 3.5 Consumed active power as a function of time when a voltage step of magnitude

–10% is applied to an incandescent lamp. As can be seen from the diagram, incandescent

lamps have no dynamic behaviour at all.

Type Switch-offvoltage

Active power before switch-off

Fluorescent lamp (2 x 36 W) ~ 160 V ~ 30 W

Mercury lamp (250 W) ~ 180 V ~ 165 W

High pressure sodium lamp (250 W) ~ 180 V ~ 180 W

Low pressure sodium lamp (91 W) ~ 80 V ~ 60 W

Low-energy lamp (11 W) ~ 40–80 V ~ 2.6 W


23

Figure 3.6 Consumed active and reactive power as a function of time when a voltage step

of magnitude –10% is applied to a mercury lamp. As can be seen from the diagram, mercury

lamps have a slight dynamic voltage dependence in reactive power and no dynamic voltage

dependence in active power.

Figure 3.7 Consumed active and reactive power as a function of time when a voltage step

of magnitude –10% is applied to a high pressure sodium lamp. As can be seen in the diagram,

high pressure sodium lamps have a clear dynamic voltage dependence in both active and

reactive power.


24

3.6.4 Accuracy analysis

As mentioned before, it is the bias in the measurement devices, which most

influences the absolute error. Figure 3.9 and Figure 3.10 show the inaccuracy in

active power and voltage for a high pressure sodium lamp and a fluorescent lamp

respectively. The crosses show the maximum limits of the absolute error for the

active power and voltage according to:

Figure 3.8 The maximum limit of the absolute error, where Pm and Um are measured

active power and voltage, and Ep, Eu respectively are the calculated maximum limits of the

absolute error.

Figure 3.9 The accuracy-variations in measured active power and voltage of a high

pressure sodium lamp. The accuracy in active power is very high, practically no vertical lines

in the “crosses” can be seen. The dominating inaccuracy, which however is small, is in the

voltage (the horizontal lines in the “crosses”).

Um–Eu Um+EuUm

Pm–Ep

Pm+Ep

Pm


25

Figure 3.10 The accuracy-variations in measured active power and voltage of a fluorescent

lamp. The accuracy in active power is very high, practically no vertical lines in the “crosses”

can be seen. The dominating inaccuracy, which however is small, is in the voltage (the

horizontal lines in the “crosses”).

There is a very small error in measured active power for discharge lamps andincandescent lamps. This is due to the error in measured active power beingdependent on the power factor of the test object, i.e. the accuracy in measured activepower increases as the power factor increases1. Therefore, if the lamps are purelyresistive or the lamps are totally compensated, the inaccuracy in active power willbe minimal.

The accuracy in reactive power has an inverse reaction to the power factor - theaccuracy in the calculated reactive power increases as the power factor decreases1,which results in an unacceptable accuracy in the calculated reactive power for lowpressure and high pressure sodium lamps, as they are almost totally compensatedover the entire voltage range (Figure 3.11). If, however, the error in reactive poweris compared to apparent power S instead of Q, the error will be less.

Mercury and fluorescent lamps have a strong voltage dependence in reactive power,stronger voltage dependence than the active power, i.e. the power factor varies withthe voltage. The power factor approaches one at very low voltages, which gives apoor accuracy in reactive power at lower voltages, but a better accuracy at highervoltages where the power factor is less (Figure 3.12).

1. A well-known problem in measurements concerning active and reactive power.


26

Low-energy lamps show a slightly poorer accuracy in both active and reactivepower, due to a low current and a bad power factor. See Appendix D which showsthe accuracy-variations in active and reactive power for the six various types oflamp.

Figure 3.11 The accuracy-variations in calculated reactive power and measured voltage of

a high pressure sodium lamp. The accuracy is unacceptable over the total voltage range.


a fluorescent lamp. The accuracy is unacceptable for low voltages. The “crosses” below

approximately 0.8 pu on the voltage axis are actually capacitive.


27

3.6.4.1 Harmonics generated by discharge lamps

The bias of the multimeters is only specified for sine waves, but discharge lightingis highly nonlinear and gives rise to considerable odd-ordered harmonic currents.The voltage across the tube itself illustrates clearly the non-linearity. Thewaveforms in Figure 3.14 show voltage uL across and current iL through a mercurylamp, where uL and iL are defined as follows:

Figure 3.13 Measured voltage and current.

Figure 3.14 The waveforms for the voltage uL across and the current iL to a mercury lamp

(recorded by a sampling Data Acquisition System, DAS).

Lamp

Reactor

Capacitor

i

+

u

–

+

uL

–

iL

Time

Ampl

itude


28

As can be seen in Figure 3.14, the voltage waveform across the discharge lampresembles a square-wave having sharp leading and trailing edges, which occurrepeatedly at the same angles, respectively, in each cycle. The voltage wave form istherefore very rich in odd-order harmonics over a wide frequency spectrum. Thecurrent wave form is more sinusoidal and has predominantly low-frequencycomponents. However, when the measurements were carried out on dischargelamps the circuit consisted of a lamp, reactor and capacitor (in some cases theigniter too) and are represented in the wave forms shown in Figure 3.15 (u and idefined as in Figure 3.13).

Figure 3.15 The feeding voltage u and current i of the lamp circuit. The feeding voltage

contains harmonics of higher orders, which can be seen as a ripple on the current wave form,

as the capacitor has a lower reactance for harmonics of higher orders; compare the

expression: In = j ω n C Un; ω = 2 π f, f = fundamental frequency. (Recorded by a sampling

Data Acquisition System, DAS)

A spectrum analyser was used to obtain the frequency spectrum for both the feedingvoltage and current of a mercury lamp. The result can be seen in a plotted frequencyspectrum, Figure 3.16 and Figure 3.17 where the different harmonic componentsare shown as per cent of the fundamental frequency.

Time

Ampl

itude


29

Figure 3.16 The frequency spectrum of the feeding voltage of a mercury lamp.

Figure 3.17 The frequency spectrum of the feeding current of a mercury lamp.

0

0.1

0.2

0.3

%

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0.4

1

100

Harmonic order

Per

cent

of

the

fund

amen

tal f

requ

ency

0

2.5

5

7.5

%

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

10

12.5

1

100

Harmonic order

Per

cent

of

the

fund

amen

tal f

requ

ency


30

3.6.4.2 The inaccuracy due to harmonics

As mentioned before, the accuracy for the multimeters is only specified for

sinewave inputs. The inaccuracy due to harmonics is determined by the crest factors

of the measurement devices, where the crest factor is defined as the peak value

divided by the rms-value of the quantity. For example the crest factor for a pure

sinewave equals .

At full scale the allowable crest factor for the multimeters is 4. This will do for

measurements on discharge lamps, as the wave forms of the current to these lamps

have a crest factor lesser than four. However, the current to a low-energy lamp has

a crest factor which equals ≈4 (Figure 3.18). During a similar test between the

multimeter and a sampling DAS (the Data Acquisition System, which was used in

the field measurement) the displayed value on the multimeter for the current and the

calculated rms value from the instantaneous values showed a good resemblance. In

Figure 3.18 the waveforms for a low-energy lamp are shown and in Appendix E the

corresponding waveforms for fluorescent and sodium lamps can be found.

There is no specification for the crest factor for the wattmeter, or if the accuracy is

specified for only sinewaves. However, during tests with the sampling DAS, the

displayed values on the wattmeter and the calculated active power from the

instantaneous values concurred. This implies that the accuracy of the wattmeter will

suffice for measurements on discharge and low-energy lamps.

Figure 3.18 The voltage u and current i waveforms for a low-energy lamp. The crest factor

of the current equals ≈4.

2

Time

Ampl

itude


31

3.6.4.3 The definition of reactive power under non-sinusoidal conditions

As we have now seen, the harmonics influence on the measurement accuracy onvoltage, current and active power can be considered as minimal. But in calculatingreactive power under non-sinusoidal conditions a problem arises concerning thedefinition. This is briefly discussed in the following.

The use of nonlinear loads has increased greatly, as has the interest for the definitionof reactive power under non-sinusoidal conditions. Two major models dominatetoday´s approach to the definition and components of the reactive power: First is theschool of Budeanu, which is sanctioned in the ANSI/IEEE Standard 100–1977.Second is the school of Fryze which influenced the International ElectricalCommission´s position. Articles in journals show the ongoing struggle to recogniseand produce a practical model acceptable to the electric utility [1, 2, 3, 4, 5]. Allsupporters of both the schools agree however, that none of the definitions arecomplete. Here follows a short presentation of the two schools, which are oftenreferred to as the frequency-domain approach and the time-domain approachrespectively:

Budeanu´s definition:

Fryze´s definition: The source current is divided into two orthogonal components,ia and ir, called “active current” and “reactive current” asfollows:

;

where u = source voltage

And with the following relations between rms values

which yields

;

To give the problem a more practical approach, look at the voltage and current of alow-energy lamp in Figure 3.18. What about the reactive power: is the lampinductive/capacitive or is it resistive? One can safely ascertain that the power factor,which is universally defined as P/S, where P is the average power andS = Irms · Urms, equals approximately 0.5.

Q UnInsinϕnn 1=

∞

∑=

i ia ir+= iaP

U2

-------u=

I2

Ia2

Ir2

+=

S2

P2

QF2

+= QF UIr S2

P2

–= =


32

In the absence of a clear definition of reactive power under non-sinusoidalconditions, the well-known definition , was used in calculating thereactive power.

It should be noted though, that when measurements were made without acompensating capacitor and the compensated reactive power was calculated, thereactive power, compared to a measured compensated and calculated reactivepower, differed. This might however also depend on the different inaccuracies in thecalculated reactive powers.

3.6.5 The static load models

The Least Square Method (LSM) was used to derive parameters for these threemodels, described in Chapter 1:

1. Exponential:

2. Linear:

3. Polynomial:

The parameters are usually derived from the measured values, which lie between90% and 110% of the nominal voltage. If it is possible (if the lamp has not switchedoff), the third model is calculated from the datums which lie between 70% and110% of the nominal voltage.

As mentioned in Section 3.6.4 about inaccuracy, lamps with a power factor close to1.0, have a high accuracy in measured active power. The models for active powerfor discharge lamps and incandescent lamps are therefore more or less accurate. Asthe accuracy in calculated reactive power for the high- and low pressure sodiumlamps is unacceptable, it is rather needless to calculate models for the reactivepower for those two.

The models for active and reactive power for low-energy lamps may be uncertain,since the inaccuracy in the measured and calculated values is slightly high.

Q S2

P2

–=

PP0------

UU0-------

np

= QQ0-------

UU0-------

nq

=

PP0------ a0 a1

UU0-------+= Q

Q0------- b0 b1

UU0-------+=

PP0------ a0 a1

UU0------- a2

UU0-------

2+ += Q

Q0------- b0 b1

UU0------- b2

UU0-------

2+ +=


33

Table 3.2 shows the calculated parameters np and nq for the exponential models.Table 3.3 shows the calculated parameters a1 and b1 for the linear models, a0 and b0are easily calculated as 1 – a1 and 1 – b1 respectively. Table 3.4 and Table 3.5 showsome calculated parameters for the polynomial load models. Since a0 + a1 + a2 = 1,it is impossible to present the parameters in an interval as in Table 3.2 and Table3.3. Instead Table 3.4 and Table 3.5 show some examples of the polynomial models.Appendix F shows recorded quantities together with their models.

Table 3.2 The parameters for the exponential model.

Table 3.3 The parameters for the linear model.

Type np nq

Incandescent lamp 1.5–1.55 –

Fluorescent lamp 2.0–2.2 4.5–6.5

Mercury lamp 2.2–2.4 4.0–6.0

High pressure sodium lamp 2.1–2.5 –

Low pressure sodium lamp 0.3–0.5 –

Low-energy lamp 0.8–1.2 1.0–1.5

Type a1 b1

Incandescent lamp 1.5–1.55 –

Fluorescent lamp 1.9–2.1 4.5–6.5

Mercury lamp 2.2–2.4 4.0–6.0

High pressure sodium lamp 2.1–2.5 –

Low pressure sodium lamp 0.4–0.6 –

Low-energy lamp 0.8–1.2 1.0–1.5


34

Table 3.4 The parameters for the active power for the polynomial model.

Table 3.5 The parameters for reactive power for the polynomial model.

3.6.6 The dynamic behaviour

The following results were obtained from the dynamic measurement test results andinclude, if the lamp is dynamic voltage dependent, power deviation (i.e. thedifference between the stationary value and the momentarily power response aftera voltage step), settling time and time constant (if a time constant is well defined):

Incandescent, low-energy and fluorescent lamps have no dynamic behaviour at all.

Type a0 a1 a2

Incandescent lamp –0.098 0.669 0.429

–0.124 0.733 0.391

Fluorescent lamp –2.180 4.55 –1.37

–1.715 3.48 –0.765

Mercury lamp –1.28 2.23 0.05

–1.817 3.679 –0.862

High pressure sodium lamp –0.002 –0.437 1.439

–0.082 –0.273 1.355

Low pressure sodium lamp 0.98 –0.34 0.36

0.91 –0.19 0.28

Low-energy lamp 0.363 0.024 0.613

0.168 0.468 0.364

Type b0 b1 b2

Fluorescent lamp 7.58 –19.34 12.76

7.56 –19.19 12.63

Mercury lamp 3.454 –9.964 7.51

2.78 –8.47 6.69

Low-energy lamp –0.009 0.654 0.355

–0.018 0.68 0.338


35

Mercury lamps have no dynamic voltage dependence in active power, but a voltagedependence in reactive power, with a reactive power deviation of approximately 5%after a voltage step, and a settling time of approximately 1–1.5 minutes.

Both types of sodium lamps have a dynamic voltage dependence in active andreactive power. High pressure sodium lamps have an active power deviation ofapproximately 5% after a voltage step with a settling time of approximately 30–60seconds, and the reactive power has a time constant of approximately 1 minute anda settling time of 2–3 minutes. Low pressure sodium lamps have approximately thesame settling time in active and reactive power ~5 minutes, and the active powerhas a power deviation of approximately 5%. Due to the insufficient accuracy incalculated reactive power for sodium lamps, it is not possible to determine thereactive power deviation.

3.6.7 Discussion

Some conclusions can be drawn from the results concerning active powerconsumption, reactive power consumption and the switch-off voltage:

Active power As can be seen in Table 3.2 to 3.5, discharge lamps have anapproximate quadratic voltage dependence, except lowpressure sodium lamp which acts as constant power load. Theincandescent lamps have a voltage dependence which isbetween constant current and constant impedance load. Low-energy lamps act as constant current load.

Reactive power Fluorescent lamps and mercury lamps have a strong voltagedependence. The other lamps have a nearly linear voltagedependence or nearly no voltage dependence at all.

Switch-off voltage The switch-off voltages for the mercury lamps and the highpressure lamps are rather high, which means for example thatlamps at the end of a street lighting circuit may switch-off evenif the feeding voltage is fairly close to nominal value.Göteborgs Energiverk experienced this, when they connecteda balancing transformer at the feeding point in order to depressthe voltage and save energy. This resulted in problems withflickering lamps and unlit lamps at the end of the lines.

The lamps have no pronounced dynamic behaviour. No differences between old andnew lamps were observed. If there was a variation, it seemed to be due to thevariations in individual test objects.


36

3.7 Laboratory measurements on electric heating devices

Up to now electric heating devices have been considered as constant impedanceloads. However, the device comprises not only a heat generating component (aresistance), but of a thermostat too. In order to investigate the influence of thethermostat, laboratory measurements were performed as a part of masters degreethesis at the Department of Electrical Power Systems, Chalmers [6, 7].

3.7.1 Test objects

The measurements for home heating appliances were taken using radiators with twodifferent types of thermostat.

1. Bimetallic regulated heating device

2. Electronic regulated heating device

Type 1 has a bimetallic thermostat and is, in Sweden, the most commonly usedtemperature regulated radiator. In Figure 3.19 the bimetallic regulated radiator isshown schematically.

Figure 3.19 A schematic diagram of the bimetallic regulated radiator.

In the bimetallic thermostat there are, apart from the bimetallic switch, anaccelerating and a compensating element. The purpose of these elements is to givea constant room temperature that does not change with the working-point of theradiator, i.e. change with the mean power output. The principal of the thermostat isas follows:

Bimetallicswitch

OFF

ON

Compensatingelement

Ceramic case

Acceleratingelement

Heatingcomponent

230 V


37

The accelerating element, which is a resistance, emits heat to the bimetallic switch

during the on-period in order to get an earlier switch-off, since the radiator

continues to emit the accumulated energy after the switch-off. This inhibits an

overshoot in room temperature that otherwise would take place, and results in

shorter on/off cycles (the time when the radiator is on, plus the time the radiator is

off) and a more steady room temperature.

The disadvantage with an accelerating element is that the bimetallic switch switches

off at different temperatures, depending on the working-point of the radiator. To

counteract this drift, a compensating element, which is a resistance with a ceramic

case, is connected to the feeding voltage during the switch-off period. Due to the

heat-accumulating ceramic case, the compensating element will emit heat

continuously during a whole on/off cycle. The compensating element reaches a

higher temperature and emits more heat the longer it is connected, i.e.the shorter the

radiator is switched on. The compensating element affects the cycle in an opposite

direction to the influence of the accelerating element, and the two components

together will give the bimetallic switch a constant over temperature. Bimetallic

regulated heating devices have a rather large variation in on/off cycles, from around

50 to 400 seconds.

Type 2 has an electronic thermostat and its main components are a thermal sensor,

a potentiometer and a control circuit together with a TRIAC power switch, see

Figure 3.20:

Figure 3.20 A schematic diagram of the electronic thermostat.

Heating

component

Triac

230 V

Comparator

Control circuit

–+

Amplifier

E

R

RThermalsensor(NTC)

Presettemperature


38

The voltage over the potentiometer - which depends on the preset temperature onthe radiator, and the voltage over the thermal sensor - which depends on the roomtemperature, are compared in the control circuit. When the conditions for reactionare reached, the control circuit gives an impulse to the TRIAC in order to switchon/off the radiator.

The electronic thermostat has a constant on/off cycle. The two radiators withelectronic thermostats which were tested, had on/off cycles of 48 and 52 secondsrespectively.

3.7.2 The climate box

A large box was built in order both to control the thermal flow and to achieveexperimental reproducibility (not in order to represent an actual room) see Figure3.21.

Figure 3.21 The climate box seen from above.

L · W · H: 2400 · 1000 · 1200 mm.

Framework: 12 mm thick plywood plates.

Insulation: 100 mm thick polystyrene.

Air flow distributor: Perforated plate with 9.5 mm quadratic holes, 52% air.

Fan: Canal ventilator.

The box was built of 12 mm thick plywood sheets and insulated on all six surfaceswith 100 mm thick polystyrene, which gives approximately - at a 5 degree deviationbetween the inside and the outside temperature - an energy loss of 30 W, which maybe considered negligible.

Test object, radiator

Perforated plate

1000

2400

Canal ventilator temp 1 temp 2


39

Holes were made on both the short sides, in order to obtain an airflow through the

box. A canal ventilator was mounted over one of the holes whilst the other hole

served as the outflow. To prevent the outflow air from influencing the inflow, the

ventilated air was blown into an adjacent room. A perforated plate was placed inside

the box to distribute the air flow. Thermometers were placed at the inflow, the

outflow and inside the box.

3.7.3 Results

The results of the measurements are illustrated in two different graphs for each

radiator. One graph shows the on-and-off periods and the other shows the mean

power which is calculated as the average power during an on/off cycle according to:

(3.3)

In Figure 3.22 and Figure 3.23, a voltage step of magnitude –20% has been applied

to the bimetallic regulated radiator. The on-and-off periods for the radiator can be

seen in Figure 3.22 and the mean-power in Figure 3.23.

Figure 3.22 On-and-off periods for a radiator with a bimetallic thermostat. The disturbance

is a voltage-step of magnitude –20%.

Pmean

P ton⋅ton toff+---------------------=

AAAAAAAAAAAA

Tim

e (s

)

800

600

400

200

0

AOn, before avoltage step

Off

On, after avoltage step

On-and-off periods


40

Figure 3.23 The mean-power for a radiator with a bimetallic thermostat. The disturbance is

a voltage-step of magnitude –20%.

Figure 3.24 and Figure 3.25 show the on-and-off periods and the mean-power,respectively, after a voltage step of magnitude –20% has been applied to theelectronic regulated radiator.

Figure 3.24 The on-and-off periods for a radiator with an electronic thermostat. The

disturbance is a voltage-step of magnitude –20%.

200

300

400

500

0 1000 2000 3000 4000 5000

Mea

n Po

wer

(W

)

Time (s)

voltage-step

AAAAA

Tim

e (s

)

40

30

20

10

0

AOn, before avoltage step

OffOn, after avoltage step

AAAAA

AAAAA

AAAAA

AAA

On-and-off periods


41

Figure 3.25 The mean-power for a radiator with an electronic thermostat. The disturbance

is a voltage-step of magnitude –20%.

3.7.4 Load model

A load model of the instantaneous value of the power consumption for one singleradiator is not useful as the radiator will go on and off in response to the thermostat.It is more appropriate to study the mean power which gives a good indication of thepower consumption over a period of time. A group of radiators will have a meanpower curve that resembles the one for a single radiator, but due to the various on-and-off times for radiators, the time constant will be different.

3.7.5 Discussion

If the radiator is within an on-period and a voltage change is applied to it, thethermostat shows no tendency to switch off momentarily, and the radiator behavesas a constant impedance load during a time period of tens of seconds regardless ofthe type of thermostat.

After this period of time the radiators begin to act dynamically. The two regulatedradiators show a difference in dynamic behaviour:

The on/off cycles of the bimetallic regulated radiator are altered immediately afterthe disturbance. The mean power is thus changed in an opposite direction to thevoltage, i.e. when the voltage step is negative the mean-power increases and vice

200

300

400

500

0 500 1000 1500 2000 2500

Mea

n Po

wer

(W

)

Time (s)

voltage-step


42

versa. The original mean power as it was prior to the disturbance is not regained.This unexpected effect is caused by the accelerating and compensating elements, asboth react as a result of the voltage change.

The size of the deviation between the mean power before and after the voltagechange depends on the working-point of the radiator and the size of the voltage step:

For example, after an increase in voltage, the effect could be that thebimetallic element will switch off the radiator and stay that way, as thecompensating element will emit too much heat to the bimetallic switch. Theinverse may be observed during a decrease in voltage, as the acceleratingelement will emit too little heat to get a switch-off, and the radiator will stayswitched on. This was both observed in the laboratory and at complementarytests at the home of one of the authors (the room temperature was changedaccordingly). On the contrary, if a decrease in voltage occurs for example ona cold day during winter when the radiator has a high working-point (theradiators are on during nearly the whole cycle), the reaction to a smalldecrease in voltage could be that the mean power stays the same, and a largerdecrease in voltage, that the mean power decreases, instead of increases.

The on/off cycles of the electronic regulated device remain the same directly aftera disturbance, and due to the change in supply voltage, the mean power is changedaccordingly. Note that for this type of regulator, the time for an on/off cycle isconstant. The temperature regulator adjusts the on-and-off times within the constantperiod and after a damped oscillation the mean power reaches approximately thesame value as before the disturbance. On cold winter days, the same applies for theelectronic regulator as for the bimetallic regulator, that the mean power maydecrease after a negative voltage step, if the working-point of the regulator is highand the voltage step is large enough.

3.8 Laboratory measurements on refrigerators

This section deals with the laboratory measurements on refrigerators. Both staticand dynamic voltage tests are presented and parameters for different load modelsfor each refrigerator are also derived. In the per-unit diagrams presented in thissection, P0, Q0 and U0 are used as base-values, where P0 and Q0 aremeasured/calculated powers at nominal voltage U0 (~230 V). The load modelsshould be applicable in the time range of minutes, not accounting for fast transientbehaviour. In addition to the static and dynamic tests a critical voltage wasobserved.


43

3.8.1 Test objects

Two refrigerators were tested:

1. Refrigerator A 155 litres

2. Refrigerator B 154 litres

Both refrigerators were new, and refrigerator A was equipped with a small freezingcompartment.

As the on/off cycle is very long, more than one hour, the influence of the thermostatwas not taken into account. The thermostat was on maximum level so that thecompressor would be running during the measurements in each test. Therefrigerators were tested in a room with an ambient temperature of 24.5 ˚C.

3.8.2 Results from static measurements

Figure 3.26 shows the quasi-stationary voltage dependence for the active powerconsumed by refrigerator A. The results for reactive power and for refrigerator Bare shown with the derived models in Section 3.8.5 and in Appendix F. The criticalvoltage is observed in the results from the static measurements as the voltage forminimum power (Figure 3.26). The results correspond to results frommeasurements performed by A. Malmquist [8].

Figure 3.26 The quasi-stationary voltage dependence for the active power of refrigerator A.


44

3.8.3 Results from dynamic measurements

Figure 3.27 shows consumed active and reactive power and voltage as a function oftime when a voltage step of magnitude –20% is applied to refrigerator A. The restof the results are shown in Appendix C.

Figure 3.27 The dynamic voltage dependence for the active and reactive power of

refrigerator A. As can be seen from the diagram, refrigerators have no dynamic behaviour.

3.8.4 Accuracy analysis

As was previously mentioned, it is the bias in the measurement devices whichinfluences the maximum limit of the absolute error. Figure 3.28 shows theinaccuracy in active power and voltage, and Figure 3.29 the inaccuracy in reactivepower and voltage. The crosses show the maximum limits of the absolute error forthe active power and voltage according to Figure 3.8 in Section 3.6.4. As can beseen in the two figures for refrigerators, both the errors in active and reactive powercan be considered small.


45

Figure 3.28 The accuracy-variations in measured active power and voltage of a

refrigerator. The accuracy in both active power and voltage is high.


a refrigerator. The accuracy in both reactive power and voltage is high.

3.8.5 The static load models

The active and reactive powers for the refrigerators show a static behaviour after avoltage step. The quasi-stationary measurements are therefore a goodrepresentation of the voltage dependence in a range of minutes, and only staticmodels have to be derived.


46

The same static load models are derived for the refrigerators as for the lamps:

1. Exponential:

2. Linear:

3. Polynomial:

The parameters for the exponential and linear load models are derived from themeasured values, which lie between 90% and 110% of the nominal voltage, and forthe polynomial model the parameters are derived from the datums which liebetween 70% and 110% of the nominal voltage.

Table 3.6 shows the calculated parameters to the exponential models p = unp andq = unq, the calculated parameters to the linear models p = a0 + a1u and q = b0 + b1uare shown in Table 3.7 (a0 = 1 – a1, b0 = 1 – b1). The calculated parameters for thepolynomial model are divided into two tables, Table 3.8 for the active powerp = a0 + a1u + a2u2 and Table 3.9 for the reactive power q = b0 + b1u + b2u2.Appendix F shows recorded quantities together with their models.

Table 3.6 The parameters for the exponential model.

Table 3.7 The parameters for the linear model.

Type np nq

Refrigerator A 1.3 – 1.6 3.1 – 3.3

Refrigerator B 1.3 – 1.8 2.8 – 3.2

Type a1 b1

Refrigerator A 1.3 – 1.7 3.1 – 3.3

Refrigerator B 1.1 – 1.8 2.5 – 3.3

PP0------

UU0-------

np

= QQ0-------

UU0-------

nq

=

PP0------ a0 a1

UU0-------+= Q

Q0------- b0 b1

UU0-------+=

PP0------ a0 a1

UU0------- a2

UU0-------

2+ += Q

Q0------- b0 b1

UU0------- b2

UU0-------

2+ +=


47

Table 3.8 The parameters for the active power for the polynomial model.

Table 3.9 The parameters for reactive power for the polynomial model.

3.8.6 Discussion

Some conclusions can be drawn from the results concerning active powerconsumption, reactive consumption and the critical voltages:

Active power The refrigerators have a voltage dependence which is betweenconstant current and constant impedance.

Reactive power The refrigerators have a very low power factor and areinstalled completely uncompensated. The voltage dependencyfor the reactive power is rather high, nq ≈ 3.2 in the exponentialload model.

Critical voltage The critical voltage is observed in the figure for the quasi-stationary voltage dependency for the active power, Figure3.24, approximately 0.6 pu. If the voltage is lower, the powerdemand increases because the compressor stops running whilethe current is not broken [8]. If, however, the voltage isincreased from a level below the critical voltage the powerdemand will increase even more until the current is broken.

There is a negligible difference between the results of the two refrigerators and theyhave no dynamic voltage dependence.

Type a0 a1 a2

Refrigerator A 2.190 –3.955 2.765

2.360 –4.330 2.970

Refrigerator B 2.453 –4.690 3.237

2.303 –4.318 3.015

Type b0 b1 b2

Refrigerator A 2.190 –5.650 4.460

2.190 –5.640 4.450

Refrigerator B 2.023 –5.307 4.284

2.022 –5.288 4.266


48

3.9 References

[1] W. Shepherd, P. Zand, ‘Energy flow and power factor in nonsinusoidalcircuits’, Cambridge Univ. Press, 1979

[2] N. L. Kusters, W. J. M. Moore, ‘On the definition of reactive power undernonsinusoidal conditions’, IEEE Trans.Power Appl. Syst., vol PAS-99, pp.1845 - 1854, Sept/Oct. 1980

[3] C. H. Page, ‘Reactive power in nonsinusoidal situations’, IEEE Trans.Instrum. Meas., vol IM-29, pp. 420 - 423, Dec. 1980

[4] P. Filipski, ‘A new approach to reactive current and reactive powermeasurement in nonsinusoidal systems’, IEEE Trans. Instrum. Meas., volIM-29, pp. 423 - 426, Dec. 1980

[5] L. S. Czarnecki, ‘Additional discussion to ‘Reactive power under non-sinusoidal conditions’, IEEE Trans.Power App. Syst., vol PAS-102, pp.1023 - 1024, Apr. 1983

[6] J. Näslund. ‘The load-voltage dependence of home heating appliances’.Diploma thesis 91-03, Dept. of Electrical Power Systems, ChalmersUniversity of Technology, Sweden, 1991, in Swedish

[7] D. Karlsson, K. Lindén, I. Segerqvist, B. Stenborg, ‘Temporary load-voltage characteristics for voltage stability studies - Field and laboratorymeasurements’, CIGRÉ Report 38-204, Paris, Aug. 30 - Sept. 5 1992

[8] A. Malmquist, ‘Dynamics of load objects - laboratory tests’, 1985, RoyalInstitute of Technology, Sweden

Chapter 4: Field measurements

49

Chapter 4

Field measurements

4.1 Introduction

It is desirable to compare the results of the laboratory measurements with resultsfrom field measurements performed with a load similar to the ones tested in thelaboratory.

Carrying out tests in co-operation with El och Trafikteknik AB, the formerdepartment of street lighting and traffic signals at Göteborg Energi AB, gave theopportunity of doing field measurements on street-lamp networks. This chapterpresents the results of these field measurements.

Field measurements have been carried out in southern Sweden in order to derivedynamic load models [1, 2]. In these measurements an essential part of thecomposite load consisted of electrically heated households. This chapter alsopresents a short discussion on how it is possible to identify the composite load inthese field measurements with the results from the laboratory measurements.

4.2 Field measurements on street-lamp networks

The field measurements on street-lamp networks were carried out in order to verifythe static and dynamic laboratory measurements on lighting. The behaviour of onelamp, measured in the laboratory, would then be compared to the behaviour of agroup of lamps.

4.2.1 Test net

The street lamps are in separate networks and are fed from ordinary sub-stations.The networks are radial but with the possibility of reconnecting in the case of fault.


50

The tests were performed in an area with mercury lamps. Two different groups weretested, ~10 000 W and ~15 000 W rated power respectively. For comparison onesingle mercury lamp was connected in parallel with the out-going group accordingto Figure 4.2.

The street-lamp networks consist exclusively of underground cables. The networkschosen for the field measurement were fed by a balancing transformer. The purposeof this arrangement is to decrease the voltage feeding the lamps in order to saveenergy. With the balancing transformer it was possible to decrease and increase thevoltage in two steps of together about 10%, with 0.3 seconds between the steps.

4.2.2 Data acquisition system

During the field measurements, a sampling data acquisition system (DAS) was usedcontrolled by a computer. The instantaneous phase-voltages and phase-currentswere measured simultaneously through a voltage divider and a current-to-voltageconverter respectively. The data acquisition system was connected according toFigure 4.1 and the measurement arrangement as in Figure 4.2.

Figure 4.1 The data acquisition system.

Current-to-voltage converter

Current

Computer

Voltage divider

Voltage

Multiprogrammer including two A/Dconverters and a high speed memory

Mass storage device

Printer


51

Figure 4.2 Measurement arrangement.

4.2.3 Dynamic measurements

In the dynamic tests, the voltage was decreased in two steps to 90 per cent or in onestep to 98 per cent of nominal voltage, or was increased from this level back tonominal voltage in one step. The instantaneous voltages and currents weremeasured during 15 minutes, with 0.1 seconds measuring-time every second, andwith a sampling rate of 15.6 kHz. With the instantaneous values the rms values forvoltage and current were calculated according to the definition. The active power Pwas calculated by integrating the instantaneous power u · i over every cycle ofvoltage

The reactive power was calculated as

where the apparent power S was calculated as Urms · Irms.

4.2.4 Accuracy

The inaccuracy in voltage and current for the measuring device is of the same sizeas for the measuring device used in the laboratory measurement. In active andreactive powers, error due to the integration is added to the error in the measuringdevice.

Current-to-voltageconverter

To multiprogrammer

Voltagedivider

Street lampnetwork

singlemercury

lamp

Frombalancingtransformer

P1T--- u i⋅ dt

0

T

∫ =

Q S2

P2

–=


52

4.2.5 Results

Figures 4.3 to 4.8 show some results from the field measurements. Figures 4.3 to4.5 show results from the measurements on the first group (with a rated power of~10 000 W). In Figure 4.3 to 4.5 the applied voltage step is of magnitude approx.–10% (225 V down to 201 V). In this test, the voltage at the end of the line wasobserved. While the voltage in the feeding station changed from 225 V to 201 V,the voltage at the end of the line changed from 215 V down to 193 V. Just after thevoltage step one lamp at the end of the line went out and one started to flicker.During the test it was also observed that one lamp in the middle of the line went out.From the measurement test results it can be concluded that yet another lamp hadgone out. After 10 minutes all these lamps were on again.

Figure 4.3 Consumed active and reactive power in one phase as a function of time, when

a voltage step of magnitude approx. –10% (225 V down to 201 V) is applied to group one.

One lamp has gone out momentarily after the voltage step. After 5 minutes it comes on.


53



One lamp has gone out momentarily after the voltage step, and one after ~2.5 minutes. After

about 10 minutes both are on again.



3.5 minutes after the voltage step one lamp goes out and after 9 minutes it comes on.

Figure 4.6 shows when the voltage is increased from 90 per cent of nominal voltageto nominal voltage level.


54


a voltage step of magnitude approx. +10% (201 V up to 227 V) is applied to group one.

Figures 4.7 and 4.8 show some results from the tests on the second group (with arated power of ~15 000 W). Figure 4.7 shows the voltage decrease and Figure 4.8the voltage increase for one phase.


a voltage step of magnitude approx. –10% (228 V down to 204 V) is applied to group two.


55


a voltage step of magnitude approx. +10% (204 V up to 228 V) is applied to group two.

4.2.6 Discussion

The results in the field measurements are similar to the results in the laboratory

measurements. The dynamic behaviour of a group of lamps resembles the

behaviour of one lamp, exhibited in the laboratory measurements. There is a slight

dynamic voltage dependence in reactive power and none in active power (c.f.

Section 3.6.6). However, the fact that some lamps might go out due to a large

voltage step and after some minutes come on again, could be considered as a small

power recovery, with an overshoot in reactive power when a lamp comes on (Figure

4.4)

In addition, the parameters for the exponential load model, derived in the laboratory

measurements are compared to parameters derived from the tests in the field

measurements. The parameters derived in the field measurements have a higher

value (n

p

= 2.9 – 3.2, and n

q

= 4 – 7), than those derived in the laboratory

measurements (n

p

= 2.2 – 2.4, and n

q

= 4.0 – 6.0).

Both the power factor (~0.7) and the wave form of the current (not shown) indicate

that the lamps in group one are uncompensated.


56

4.3 Field measurements on composite load

Field measurements carried out in Southern Sweden [1, 2] show that the activepower recovers after a voltage step. Within five minutes, more than half of the initialactive power reduction has been recovered. This power recovery occurs in wintermeasurements, while in summer measurements there is hardly any recovery at all.

The measurements were performed in two areas with different load-composition,but in both areas an essential part of the load consists of electrically heatedhouseholds.

Out of the results of the field measurements it seems reasonable to assume that thecomposite load consists mainly of two different parts, one static part - theinstantaneous power change, and one dynamic part - the power recovery [1].

The results from the laboratory measurements can be used to identify the compositeload (Chapter 2). Non regulated radiators together with lighting, refrigerators andother non dynamic voltage-dependent loads will have an instantaneous powerchange after a step in voltage. A group of thermostat regulated radiators withbimetallic thermostats, will have an inverse reaction to voltage changes in sense ofpower and due to various on-and-off time have a slow performance (Section 3.7).These two behaviours, the instantaneous power-change and the slow inversereaction, could result in a power-recovery like the one measured in the performedfield measurements in southern Sweden.

In order to investigate the load behaviour tests similar to the field measurementsperformed in Southern Sweden have been carried out by the CEGB [3, 4]. In thesetests, the only power recovery that is mentioned is that due to automatic tapchangers.


57

4.4 References

[1] D. Karlsson, K. Lindén, I. Segerqvist, B. Stenborg, ‘Temporary load-voltage characteristics for voltage stability studies - Field and laboratorymeasurements’, CIGRÉ Report 38-204, Paris, Aug. 30 - Sept. 5 1992

[2] D. Karlsson, ‘Voltage stability simulations using detailed models based onfield measurements’, Technical Report no. 230, Dep. of Electrical PowerSystems, Chalmers University of Technology, Sweden

[3] G. Shackshaft, P. H. Ashmole, ‘The influence of load characteristics onpower system performance - A C.E.G.B viewpoint’, CIGRÉ Report 31-02,Paris, Aug. 30 - Sept. 7 1978

[4] G. Shackshaft, C. Eng, O. C. Symons, J. G. Hadwick, ‘General-purposemodel of power-system loads’ PROC. IEE, Vol 124, No 8, August 1977


58

Chapter 5: Voltage stability with regard to system and load characteristics

59

Chapter 5

Voltage stability with regard to system and load characteristics

5.1 Introduction

Incidents causing system voltage instability have occurred with increasingfrequency throughout the world. This is due to the fact that the average transmissionline is more heavily loaded than ever before. When analysing the incidents, theimportance of load characteristics and system load level are emphasized [2, 3].Curves that describe the relationship between the voltage and the transmitted activeand reactive power respectively (UP– and QU–curves) are widely used as tools forvoltage stability analysis, and are also used when planning a system [1, 4].

This part of the thesis will describe how the UP– and QU–curves together with theload characteristics can be used to show if a system, due to an incident, approachesa critical situation. The effects of two kinds of incidents are studied: an increase inthe load, and a decrease in the feeding voltage. The influence of a tap changer onboth the load characteristics and the UP– and QU–curves are also studied. Thesestudies are intended to elucidate the processes described rather than to recommendany necessary action.

5.2 System characteristics

The UP– and QU–curves for a busbar in an electric power system can be calculatedwith multiple load flow calculations or, if the system is simple enough, with powerflow equations. Figure 5.1 shows a small system of transmission lines andtransformers.


60

Figure 5.1 A small system. A load is connected to bus 3 and is of exponential type

according to the definitions in Chapter 1.

Simplify the system in Figure 5.1 by regarding bus 2 as an infinite bus. The systemis then a single line represented with an impedance Z

L

according to Figure 5.2.

Figure 5.2 Model of a single line. E and U represent the absolute value of the voltages, and

ψ

=

ψ

2 -

ψ

3 is the angle between the voltages. P and Q represent the transmitted active and

reactive power.

The power flow equations for the receiving end in this system can be expressed as

(5.1)

(5.2)

where the loss-angle

If the variables in the power flow equations are changed to per unit values with E asbase-voltage and Z

L

as base-impedance giving E

2

/Z

L

as base-power, and thenrearranged (Appendix G) we get

(5.3)

1

2 3

E U

P / Q

Z = R +jXL L Lψ

3ψ

2

PU

2

ZL------- sin δ –

EUZ

L

-------- sin ψ δ + ( ) +=

QU

2

ZL------- cos δ –

EUZ

L

-------- cos ψ δ + ( ) +=

δ arctanRL

XL-------=

p2

u2

1 u2

– 2 p sin δ q cos δ + ( ) – [ ] q 2 –=


61

For simplification the resistance will be neglected, the loss angle

δ

is then zero,which can be an acceptable approximation if the voltage level is 130 kV or higher.Equation (5.3) then becomes

(5.4)

The equations for the up– and qu–curves are achieved by isolating u and q,respectively (Appendix G)

(5.5)

(5.6)

In the equations for the up–curve (Equation (5.5)) and qu–curve (Equation (5.6)) theactive power p or the reactive power q respectively may be treated as a constant oras an expression of the other. A constant p or q gives the opportunity to studycontrol issues. If for instance the reactive power q is drawn as a function of thevoltage u with the active power p as a constant it is possible to study how to controlthe voltage by regulating the reactive power. When studying stability issues it isperhaps more appropriate to have p or q as a function of the other. What is theninteresting to study is the transmission possibilities at different load situations. Withthe up– or qu–curve in hand it is possible to observe if a working-point, i.e. asolution of a power flow calculation, is voltage stable [1].

The up–curves show the relationship between voltage and active power for a givenreactive power or with the reactive power q as a function of the active power p,q = f(p). The most commonly used function is q = p · tan

ϕ

with a given tan

ϕ

. Figure5.3 shows three up–curves for given tan

ϕ

, tan

ϕ

= 0.5, 0 and –0.5. In these curves aworking-point is considered to be voltage stable if it is located on the part of the up–curve that is above the curve that gives maximum transmitted power [1].

Note that in this context, maximum transmitted power does not imply that the anglebetween the voltages,

ψ

, is 90°. The dashed curve in Figure 5.3 shows p = u, thetransferred power at

ψ

= 90°. As seen in Figure 5.3 there could be a solution of apower flow that is unstable in terms of voltage but at a steady state angle less than90 degrees [1].

p2

u2

u2

q+( )2

–=

u 12--- q– 1

4--- q– p

2–±=

q u2

– u2

p2

–±=


62

Figure 5.3 The figure shows the up–curves for tan

ϕ

= 0.5, 0.0 and –0.5. The dotted curve

connects the points for maximum transmitted power pmax, where dp/du = 0. The dashed

curve shows p = u, i.e.

ψ

= 90°.

The qu–curves show the relationship between the reactive power and voltage forgiven active power or with the active power p as a function of the reactive power q,p = f(q), where the former alternative is the most commonly used. Figure 5.4 showsthree qu–curves with p as a constant, p = 0.25, 0.5 and 0.75, and Figure 5.5 showsthe qu–curve with p = q/tan

ϕ

, tan

ϕ

= 0.5 and –0.5. In the qu–curve a working-pointis considered voltage stable if it is on the right part of the curve, to the right of theminimum-value when p is constant (Figure 5.4) and to the right of the maximum-and the minimum-point when p = q/tan

ϕ

(Figure 5.5).

Figure 5.4 The figure shows the qu–curves for p = 0.25, 0.5 and 0.75. The dotted curve

connects the points where dq/du = 0. The dashed curve shows q when

ψ

= 90°. Note that the

q–axis is negative.

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1

1.2

0.50

–0.5= 0

ψ = 90˚

dpdu1.2

1.0

0.8

0.6

0.4

0.2

u

p

0.2 0.4 0.6 0.8 1 1.2 1.4

0.8

0.6

0.4

0.2

0

-0.2

ψ = 90˚

0.750.50.25–0.8

–0.6

–0.4

–0.2

0

0.2

u

qdq

du= 0


63

Figure 5.5 The figure shows the qu–curve for tan

ϕ

= 0.5 and –0.5. The dashed curve

shows q when

ψ

= 90°. The dotted curve connects the points where dq/du = 0. Note that the

q–axis is negative.

5.3 Load characteristics

To study the impact of various load characteristics on voltage stability, the loadcharacteristics are drawn together with the up- and qu–curve. The intersection of theup- or qu–curve and the load characteristic will be the working-point of the system,the solution of a power flow calculation. The power demand is described by anexponential load model (see Chapter 1), according to Equation (5.7) and Equation(5.8)

(5.7)

(5.8)

where n

p

and n

q

is the sensibility to voltage, u is the actual voltage feeding the loadand p

0

and q

0

is the rated power at nominal voltage u

0

, here defined as the voltageat the initial working-point. Note that u, u

0

, p, p

0

, q, q

0

are in per unit with the samebase-values as for the power flow equations. With various n

p

and n

q

different loadcharacteristics are obtained.

0.2 0.4 0.6 0.8 1

0.4

0.2

0

ψ = 90˚ = 0dqdu

u

q

–0.4

–0.2

0

tanϕ = –0.5

tanϕ = 0.5

p p0uu0-----

np

=

q q0uu0-----

nq

=


64

5.4 Tap changer influence on load characteristics

The power demand, Equations (5.7) and (5.8), is a function of the voltage feedingthe load. It is therefore of interest to study different load characteristics togetherwith the action of a tap changer that regulates the voltage feeding the load. To dothis, the system in Figure 5.2 is increased at the receiving end with an idealtransformer equipped with an on-load tap changer (Figure 5.6).

Figure 5.6 An ideal transformer with a tap changer, marked here with an unfilled arrow,

added to the small system.

In this study the voltage u will be referred to as the voltage at the feeding side of thetransformer and u′ as the voltage at the load side. The power demand as a functionof the voltage u at the feeding side of the transformer and the tap changer position,is expressed as

(5.9)

(5.10)

Two kinds of incidents are observed. An increase in load and a decrease in feedingvoltage. For both disturbances the influence of a tap changer is studied as the tapchanger automatically works to keep the voltage u´ at a specified value, here thevoltage at the initial working-point.

5.4.1 Increase in load

The increase in load is achieved by increasing the rated power p0 and q0 inEquations (5.9) and (5.10). This could illustrate an increase in the amount ofconnected load devices.

e ψ2 zL (1+∆):1u ψ3 u′ ψ3

p/q

p p0

u1 ∆+-------------

u0-------------

npp0

u0

np------- 1

1 ∆+( )np

----------------------unp= =

q q0

u1 ∆+-------------

u0-------------

nqq0

u0

nq------- 1

1 ∆+( )nq

----------------------unq= =


65

Use Equation (5.5) to draw the up–curve. If tanϕ is assumed constant, i.e.q = p · tanϕ, Equation (5.5) is changed to

(5.11)

Figure 5.7 shows the up–curve (Equation (5.11)) and three different loadcharacteristics (Equation (5.9)) with np = 0, 1 and 2, together with a 40% increasein load for the different load characteristics.

Figure 5.7 The up–curve for tanϕ = 0.5 is shown together with the three load

characteristics, np = 0, 1 and 2, before and after a 40% increase in load. The continuous load

characteristics show the situation before the increase in load and the dotted load

characteristics show the situation after the increase in load but before any tap changer action.

In this instance, after the increase in load there will be three different working-points with positions dependent on the load characteristics (Figure 5.7). A constantload characteristic will give the highest power demand, and the higher voltagedependency the lower power demand.

As the voltage in the receiving end of the system decreases due to the increase inload, the tap changer will act to restore the voltage on the load-side of thetransformer. The action results in a new ∆ in Equation (5.9), where ∆ is decided bythe tap changer position. On condition that there is a continuous tap changer thevoltage at the load-side of the transformer is restored to the specified value. Figure5.8 shows how the action of the tap changer increases the load when restoring thevoltage on the load-side of the transformer.

u 12--- p tan ϕ – 1

4--- p tan ϕ – p

2 – ± =

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1

np=0 np=1 np=2

p

u

tanϕ = 0.5


66

Figure 5.8 The figure shows how the tap changer increases the load with np = 2 seen from

the feeding side of the transformer when restoring the voltage at the load side. The continuous

load characteristic show the initial stage and the dotted curves show when the load is

increased and how it is increased even more by the tap changer action.

However a tap changer is not continuous, the voltage is regulated in discrete steps.A normal step-size is 1.67% and the number of possible steps is usually 9 in eachdirection from nominal position.

The above study can also be made in a qu–curve. Equation (5.6) with p = q/tan

ϕ

andtan

ϕ

as a constant, i.e. Equation (5.12), is the expression for the qu–curve withconstant tan

ϕ

(5.12)

However, drawing the qu–curve with the active power p as a function of the reactivepower q might seem a little strange. It implies that the reactive power can bechanged and that the active power will follow according to the function, when inpractice it is often the other way round. It might, however, be of interest to studyhow the reactive power q varies when the active power changes. Showing this in aqu–curve with p as a constant together with the load characteristics results in a newqu–curve as well as a new load characteristic every time the tap changer acts. InFigure 5.9 the qu–curve with tan

ϕ

= 0.5 is drawn together with three loadcharacteristics, n

q

= 0, 1 and 2 in Equation (5.10), and a 40% increase in load forthe different load characteristics. As tan

ϕ

is constant, q

0

in Equation (5.10) is equalto p

0

· tan

ϕ

. Henceforth the studies are concentrated upon up–curves but some qu–curves will be presented for the sake of completeness.

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1

p

u

tan

ϕ = 0.5

np = 2

q u2

1 1

tan2ϕ

--------------+-----------------------– u

4

1 1

tan2ϕ

--------------+ 2------------------------------- u

2u

4–

1 1

tan2ϕ

--------------+-----------------------+±=


67

Figure 5.9 QU-curve, tan

ϕ

= 0.5, together with three load characteristics, nq = 0, 1, and 2,

with a 40% increase in load. Note that the q–axis is negative.

5.4.2 Decrease in feeding voltage

When changing the feeding voltage E to a new value E

new

the equations for the UP–and QU–curves change (Appendix G) and, in the case of constant tan

ϕ

, Equations(5.11) and (5.12) change according to Equations (5.13) and (5.14).

(5.13)

(5.14)

The decrease in feeding voltage results in a new up- or qu–curve (Figure 5.10 andFigure 5.11) and together with the load characteristics there is a new working-point.

0.2 0.4 0.6 0.8 1

-0.05

-0.1

-0.15

nq=0nq=1nq=2

u

q

0.05

0.1

0.15 tan

ϕ = 0.5

u12---enew

2p tan ϕ –

14---e new

4 e new

2 p tan ϕ – p

2 – ± =

q u2

1 1

tan2ϕ

--------------+-----------------------– u

4

1 1

tan2ϕ

--------------+ 2-------------------------------

u2enew

2u

4–

1 1

tan2ϕ

--------------+-----------------------------+±=


68

Figure 5.10 A decrease in feeding voltage results in a new up–curve. The figure shows the

original up–curve and the up–curve for a reduced feeding voltage, both with constant tan

ϕ

,

together with three load characteristics, np = 0, 1 and 2 in Equation (5.9).

Figure 5.11 The decrease in feeding voltage illustrated with two qu–curves together with

three load characteristics, nq = 0, 1 and 2 in Equation (5.10).

As the feeding voltage is reduced, so is the load voltage. The tap changer will thenoperate to restore the voltage on the load side of the transformer as in the case withincrease in load. Figure 5.12 shows a decrease in feeding-voltage together with theload characteristics for n

p

= 2 in Equation (5.9). With a continuous and infinite tapchanger the voltage on the load side of the transformer will be restored to thespecified value.

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1

enew = 1

enew = 0.8

p

u

np = 0 1 2

0.2 0.4 0.6 0.8 1

0.4

0.2

0

enew = 1

enew = 0.8

q

u

nq = 012

–0.4

–0.2

0


69

Figure 5.12 Two up–curves describing a 20% decrease in feeding voltage together with

load characteristics, np = 2. The tap changer increases the load seen from the line-side as it

restores the voltage-level on the load side of the transformer.

5.4.3 Discussion

5.4.3.1 Increase in load

The use of UP– and QU–curves, together with load characteristics, show that if aload is voltage-dependent a tap changer connected as per Figure 5.6 will increasethe power demand when restoring the voltage feeding the load. In the casespreviously outlined, where the increase in load could be described as an

increase inthe amount of connected load devices or rather an increase in rated power

, the tapchanger tends to give a voltage-dependent load a characteristic of a “constantpower” -load.

If instead the increase in load is described as an

increase in actual power demand

,i.e. from a given working-point with an arbitrary load characteristic the load-demand is increased to a new working-point with a load characteristic that is notnecessarily the same as the one in the first working-point (Figure 5.13). Then thetap changer action will increase the power demand in the case of a voltage-dependent load when restoring the voltage feeding the load, resulting in a higherpower demand than if the load characteristic had been “constant power” (Figure5.14). In Figure 5.13 the up–curve for tan

ϕ

= 0.5 is drawn with different loadcharacteristics. The actual power demand is increased by 40% and the loadcharacteristics show the state before any tap changer action. Figure 5.14 shows theup–curve and the load characteristics when the tap changer has restored the voltage

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1

enew = 1

enew = 0.8

p

unp = 2

→ → →


70

on the load-side of the transformer. A high voltage dependency tends to move theoperation-point towards a critical situation.

Figure 5.13 UP–curve for tan

ϕ

= 0.5 together with four load characteristics and a 40%

increase in actual power demand.

Figure 5.14 UP–curve for tan

ϕ

= 0.5 with three load characteristics when the tap changer

has restored the load-voltage after an increase in load.

The above results are all valid when tan

ϕ

is positive, or if tan

ϕ

is negative - theworking-point is on the upper half and to the right of maximum voltage on the up–curve (Figure 5.15). These are the normal working conditions.

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1

p

unp=0 np=1 np=2 np=3

tanϕ = 0.5

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1

p

u

np=0 np=1np=2

np=3

tanϕ = 0.5


71

Figure 5.15 The up–curves for tan

ϕ = 0.5 and –0.5. Note that when tanϕ is negative the

maximum voltage is not equal to the voltage for the system in no-load running.

A moderate increase in load however requires a high-load situation for the systemto turn into a voltage collapse.

5.4.3.2 Decrease in feeding voltage

When there is a decrease in feeding voltage it is of course an advantage if the loadis voltage dependent, the more so the better, as a voltage dependent load willdecrease as the voltage decreases. The action of a tap changer to restore the load-voltage will however increase a voltage dependent load, and if the decrease involtage is not greater than the control-range of the tap changer, the voltage on theload side of the transformer will be restored to its original value and the load couldbe considered constant (Figure 5.16).

Figure 5.16 A 10% decrease in feeding voltage. In seven steps the tap changer has restored

the voltage on the load side of the transformer, and the working-point of the two voltage-

dependent load characteristics are moved to the one of the constant-power characteristic.

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1

1.2

p

u

tanϕ = –0.5

tanϕ = 0.5

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1

p

u

enew = 1

enew = 0.9

np=0 1 2


72

5.5 Tap changer influence on system characteristic

The single line dealt with earlier in this chapter is in reality a part of a large systemwith the possibilities of regulating system voltages. Recall the system in Figure 5.1,and represent the part to the left of bus 2 by an equivalent “infinite” voltage behindan impedance and an ideal transformer regulating the voltage feeding the lineaccording to Figure 5.17.

Figure 5.17 A system with an equivalent voltage Ei behind an impedance Zi both reduced

over the nominal ratio of the transformer, and a tap changer regulating the voltage E.

The tap changer is assumed to regulate voltage E to a value equal to the open circuitvoltage for the system, with the tap changer in position n = 1:1, here called the initialvoltage. Two incidents are studied: an increase in load and a decrease in feedingvoltage Ei′. In this study only the tap changer regulating the voltage E is taken intoconsideration, the voltage U is not regulated. The studies in Section 5.4 (regulatingvoltage U) might, however, just as well be applied to the more general system in thissection as to the single-line system in Section 5.4. The load connected to the systemis of the exponential model presented in Equation (5.7).

The equation for the up–curve for the system in Figure 5.17, with Ei′ as base-voltageand ZL as base-impedance, can be written as (Appendix H)

(5.15)

where and

and , is the total impedance of the system in per unit, i.e.

(5.16)

Note that δ is the loss-angle for the entire system.

Z′i(1+∆i):1

P/QZL

1 2 3

E′i E U

u

1

ni2

----- 2zA–

2----------------------

1

ni2

----- 2zA– 2

4------------------------------ z

2p

2q

2+( )–±=

ni 1 ∆i+= A psinδ qcosδ+=

z ZZL------=

ZZi

1 ∆i+( )2---------------------- ZL+=


73

As voltage E is regulated to the open circuit value of the system, it is possible todraw the up–curves for both the entire system and for the single line to the right ofbus 2 in the same chart (Figure 5.18).

Henceforth the resistance will be ignored. If the reactance Xi is written as α · XLwhere α is a constant, the total impedance of the system is written as

(5.17)

Assume q = p · tanϕ, where tanϕ is constant. The equation for the up–curve for theright part of the system, the single line, is equal to Equation (5.11), and for the entiresystem the equation is equal to

(5.18)

where

Equations (5.11) and (5.18) are graphically presented in Figure 5.18 with tanϕ = 0.5,α = 1 and ∆i = 0.

Figure 5.18 The continuous curve in the figure is the up–curve for the entire system. The

dotted curve is the up–curve for the single line to the right of bus two.

X XLα

1 ∆i+( )2---------------------- 1+=

u

1

ni2

----- 2 α

ni2

----- 1+

ptanϕ–

2-------------------------------------------------- 1

4ni4

---------1

ni2

----- α

ni2

----- 1+

ptanϕ– α

ni2

----- 1+ 2

p2

–±=

ni 1 ∆i+=

0.05 0.1 0.15 0.2 0.25

0.2

0.4

0.6

0.8

1

p

u

(5.18) (5.11)


74

5.5.1 Increase in load

Assume that a load with a constant power characteristic is connected to the system

at bus three and that there will be an increase in load. The various up–curves in

Figure 5.19 show how the up–curve for the entire system changes in accordance

with the tap changer position.

Figure 5.19 The figure shows an increase in load and how the up–curve for the entire

system changes as the tap changer restores the voltage at bus two to the initial value. The

dotted up–curve is the up–curve for the single line in the right part of the system in Figure

5.17. The dotted up–curve is only valid when the voltage E at bus two is at the initial value.

When the load is connected, working-point 1 is the intersection of the load

characteristic and up–curve a (up–curve for the entire system before any tap

changer action, Figure 5.19). The tap changer restores the voltage at bus 2 to its

initial value and up–curve d (the up–curve for the single line when the voltage E is

at the initial value) is valid. As the tap changer action results in a new ∆i there will

be a new up–curve for the entire system (up–curve b) and working-point 2 (the

intersection of the load characteristic, up–curve b and up–curve d) becomes valid.

After the increase in load, the new working-point is at the intersection of the current

up–curve for the entire system, up–curve b, and the new load characteristic, i.e.

working-point 3. If the tap changer is able to restore the voltage at bus 2 to its initial

value, the up–curve for the entire system is up–curve c. Also up–curve d is valid for

the single line and the final working-point will be number 4 in Figure 5.19.

Note that it is the voltage at bus 3 that is shown in the up–curves in Figure 5.19.

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1a b c d

21

43

p

u

np = 0


75

The scenario described above, but now with a load characteristic of type “constantcurrent” for the connected load, is shown in Figure 5.20. The working-points arenumbered in order of appearance.

Figure 5.20 The same scenario as that shown in Figure 5.19 but here with a connected load

of “constant current” -type.

5.5.2 Decrease in feeding voltage

When changing the feeding voltage E′i to a new value E′i(new) Equation (5.18)changes according to Equation (5.19) (Appendix H)

(5.19)

where and

Assume a load with a characteristic of type “constant power” connected to thesystem at bus three and that there will be a decrease in feeding voltage. Figure 5.21shows the up–curves for the different stages in the scenario.

The initial stage, when the load is connected but before any tap changer action, isshown by up–curve a, and the intersection of this curve and the load characteristicis the current working-point, (number 1 in Figure 5.21). The tap changer restoresthe voltage at bus 2 and changes the up–curve from curve a to curve b. Moreover,

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1a b c d

21

43

p

u

u

e′i(new)

ni-----------------

2

2A–

2---------------------------------------

14---

e′i(new)

ni-----------------

4 e′i(new)

ni-----------------

2

A– α

ni2

----- 1+ 2

p2

–±=

ni 1 ∆i+= A α

ni2

----- 1+

ptanϕ=


76

at this stage also up–curve d (the up–curve for the single line) is valid since thevoltage at bus 2 is at the initial value, and the working-point is at the intersection ofthe load characteristic and up–curves b and d (i.e. working-point number 2). Whenthe feeding voltage decreases the up–curve for the entire system changes accordingto Equation (5.19) from up–curve b to up–curve c. The working-point at this stageis working-point number 3. If the tap changer is able to restore the voltage at bus 2,up–curve e and d are valid and the system is back to working-point 2. However,observe the differences in pmax for up–curve b and e.

Figure 5.21 The figure shows the way in which the up–curve changes for a system

according to Figure 5.17. This results after a decrease in feeding voltage when the tap changer

action restores the voltage at bus 2 to the open circuit value.

5.5.3 Discussion

The changes of the up–curve due to the tap changer action may be summarized inthe equation for pmax, pmax = f(E′i(new),Z,ϕ,∆i) (see Appendix H)

(5.20)

In other words, apart from the load-factor and possibly a change in feeding voltage,pmax is a function of the tap changer position.

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1 a b c d e

213

p

u

np = 0

pmax

e′i(new)

1 ∆i+-----------------

2

1z--- cosϕ

2 sin ϕ δ+( ) 1+[ ]------------------------------------------=


77

Observe that δ is the loss-angle for the entire system. If the resistance is ignored andthe impedance for the system is written according to Equation (5.17), Equation(5.20) may be simplified to

(5.21)

5.6 Reflections on some simplification

When drawing the UP– and QU–curves in the previous part, as well as the reductionof the system to a single line, two major simplifications have been made, viz theneglection of resistance and the assumption of constant tanϕ (i.e. the active andreactive power have the same voltage dependence). The influence of thesesimplifications are discussed in the following.

5.6.1 Resistance

In theoretical studies of the kind presented in this chapter most authors neglect theresistance, especially if the systems studied are regarded as transmission systems(EHV). However the up– and qu–curves are valid at any voltage level and it couldtherefore be of interest to get an overview of the effect the resistance has on the up–curves.

The size of the resistance per unit of length for a line in a power system isdetermined by the area of the phase conductors which are roughly determined bythe transmitted power, i.e. a function of the voltage level, whereas the reactance perunit of length has almost the same value on every voltage level. Therefore it ispossible to estimate an approximate ratio between the resistance and reactance foreach voltage levels according to Table 5.1.

Table 5.1 The approximated ratio between the resistance and the reactance for a

transmission line at different voltage levels, and the loss angle δ.

U, [kV] RL/XL ~δ = arctan(RL/XL), [˚]

10 2.5 68

20 1.7 59

40 0.8 39

50 0.5 27

70 0.4 22

130 0.1 6

400 0.07–0.05 4–3

pmax e′i(new)( )2 1

α 1 ∆i+( )2+

-------------------------------- cosϕ2 sinϕ 1+( )----------------------------=


78

To derive the equation for the up–curve when δ is not equal to zero, Equation (5.3)with q = p · tanϕ is solved for u (Appendix G), and the expression for the voltage uwill be according to

(5.22)

where

Figure 5.22 shows the up–curves for four different δ with correspondingRL/XL = 0.8, 0.4, 0.05 and 0.0, and with q = p · tanϕ, tanϕ = 0.5.

Figure 5.22 The up–curves for four different d with corresponding RL/XL = 0.8, 0.4, 0.05

and 0.0, and with q = p · tanϕ, tanϕ = 0.5.

With increasing RL/XL maximum transmittable power, pmax, decreases until it hitsa limit and then increases. The maximum transmittable power, pmax, as a functionof ϕ and δ, pmax = f(ϕ,δ), is expressed according to Equation (5.23) (Appendix G)

(5.23)

For a given ϕ, or tanϕ, minimum pmax is obtained for maximum denominator inEquation (5.23) i.e. δ = 90 – ϕ, which gives the limit for the different up–curves witha given tanϕ but various RL/XL ratio (Figure 5.22).

The influence of increasing RL/XL ratio can also be described by drawing the curveconnecting the points with pmax (where dp/du = 0 in the up–curve) for different tanϕ

u 12--- pB– 1

4--- p

2B

2pB– p

21 tan

2ϕ+( )2

–+±=

B sinδ tanϕcosδ+=

0.1 0.2 0.3

0.2

0.4

0.6

0.8

10.8 0.4 0.05 0

p

u

pmaxcosϕ

2 sin ϕ δ+( ) 1+( )------------------------------------------=


79

but a given RL/XL ratio. The equations in question in this case are (Appendix G) forthe lossless line, RL/XL = 0, according to

(5.24)

and for RL/XL ratios not equal to zero

(5.25)

The equations are graphically presented in Figure 5.23, where four curves withgiven δ, corresponding to RL/XL = 0.8, 0.4, 0.05, and 0.0, (c.f the dotted curve inFigure 5.3).

Figure 5.23 Curves connecting the points with pmax for various tanϕ and given δ with

corresponding RL/XL = 0.8, 0.4, 0.05, and 0.0.

When looking at the curves in Figure 5.23 it is important to have in mind that aworking-point at a voltage u greater than one per-unit needs a negative tanϕ, if theloss-angle δ is large, then tanϕ need to be much less than –1 which is hardly the factunder normal conditions.

u p=pmax( ) pmax2 1

4---+=

u p=pmax( )1 2pmaxsinδ–

2sin2δ

----------------------------------1

sinδ---------- 1

4sin2δ

----------------pmax

sinδ-----------– pmaxsinδ 1

4---–+±=

0.5 1 1.5

0.5

1

1.5

0

p

u0.05

0.4

0.8


80

5.6.2 Relation between the active and the reactive power.

The most common function describing the relationship between the active power pand the reactive power q is, q = p · tanϕ, with tanϕ as a constant. This implies thatq has the same voltage-dependency as p. With the exponential load-model used inthis chapter it is possible to derive an expression for p as a function of q and viceversa with various np and nq (Appendix I) under the conditions that the nominalvoltages u0p and u0q are equal and that the rated powers p0 and q0 are known. Forthe reactive power q the expression will be according to

(5.26)

Insert the expression in Equation (5.12) to draw the up–curve. Figure 5.24 showsthree up–curves for a loss-less line with np = 2 and nq = 1.3, 2, and 2.7.

Figure 5.24 Three up–curves with np = 2 and nq = 1.3, 2, and 2.7. q0 = p0 tanϕ where

tanϕ = 0.5.

q q0pp0-----

nq

np-----

=

0.1 0.2 0.3

0.2

0.4

0.6

0.8

1

p

u

nq = 1.3nq = 2nq = 2.7


81

5.7 References

[1] J. Falck Christensen et al, ‘Planning Against Voltage Collapse’ CIGREReport SC 38-01 TF03 (Oct 1986)

[2] K. Walve, ‘Modelling of power system components at severedisturbances’ CIGRE Report 38-18 (1986)

[3] A. Kurita, T. Sakurai, ‘The Power System Failure on July 23, 1987 inTokyo’, Proceedings of the 27th Conference on Decision and Control,Austin, Texas, December 1988, pp.2093-2097

[4] K. Clark Harrison, ‘New Challenge: Voltage Stability’, IEEE PowerEngineering Review, (April 1990)


82

Chapter 6: Future work

83

Chapter 6

Future work

Investigations on load model structures are essential, i.e. what the load modelshould include in different stability analysis, for example if tap changer action, timedependence, nonlinearities etc. are to be included. Important parts are the varioustime constants of the load objects and the time range of the studied problem.

A part of this investigation could include, for example, a further study of electricalheating in different time ranges. The electric heating viewed as an energy problem(mean power) in transient stability studies and a maximum power problem involtage stability studies (statistical approach).

Chapter 6: Future work

84

Bibliography

85

Bibliography

J. Bubenko, ‘Belastningens statiska karakteristikor’, Rapport Vattenfall nr 1285,6/10 1961

B. Stenborg, ‘Belastningskarakteristikens inverkan på den dynamiska stabiliteten’,Research report 6302, Dept. of Applied Electrical Engineering, CTH 1963, inSwedish

T. Johansson, J. Bubenko, Å. Ölwegård, ‘Belastningens representation vidstabilitetsundersökningar’, Rapport Vattenfall ( SKN-TJ/MW 8/4 1963 pr 231)

Maurice H. Kent, Wayne R. Schmus, Francis A. McCrackin, Luther M. Wheeler,‘Dynamic Modeling of Loads in Stability Studies’, IEEE Transactions on PowerApparatus and Systems, Vol. PAS-88, No 5, May 1969, 756 - 763

System Load Dynamic - Simulation Effects and Determination of Load Constants,IEEE Transactions on Power Apparatus and Systems, Vol. PAS-92, No 2, March-April 1973, 600- 609

Arne Sjöström, ‘Asynkronmotormodeller av olika belastningsobjekt’, Rapport SD-5080, Vattenfall, Stockholm, Februari 1981

Roderick J. Frowd, Robin Podmore, Mark Waldron, ‘Synthesis of Dynamic LoadModels for Stability Studies’, IEEE Transactions on Power Apparatus and Systems,Vol. PAS-101, No 1, January 1982, 127 - 135

C. Concordia, S. Ihara, ‘Load Representation in Power System Stability Studies’,IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, No 4, April1982, 969 - 977

J. R. Ribeiro, F. J. Lange, ‘A New Aggregation Method for Determining CompositeLoad Characteristics’, IEEE Transactions on Power Apparatus and Systems, Vol.PAS-101, No 8, August 1982, 2869 - 2875

S. A. Y. Sabir, D. C. Lee, ‘Dynamic Load Models derived from data acquiredduring System Transients’, IEEE Transactions on Power Apparatus and Systems,Vol. PAS-101, No 9, September 1982, 3365 - 3372

Bibliography

86

A. A. Fouad, Vijay Vittal, ‘Power System Response to a Large Disturbance’, IEEETransactions on Power Apparatus and Systems, Vol. PAS-102, No 11, November1983, 3534 - 3540

W. J. Wilson, J. D. Aplevich, ‘Dynamic Equivalent Power System Models’, IEEETransactions on Power Apparatus and Systems, Vol. PAS-102, No 12, December1983, 3753 - 3760

K. P. Wong, W. Derek Humpage, T. T. Nguyen, K. K. K. Ho, ‘Dynamic load modelsynthesis’, IEE PROCEEDINGS, Vol. 132, Pt C, No 4, July 1985, 179 - 188

T. Ohyama, A. Watanabe, K. Nishimura, S. Tsuruta, ‘Voltage dependence ofcomposite loads in power systems’, IEEE Transactions on Power Apparatus andSystems, Vol. PAS-104, No 11, November 1985, 3064 - 3073

Marc Langevin, Philippe Auriol, ‘Load response to voltage variations and dynamicstability’, IEEE Transactions on Power Systems, Vol. PWRS-1, No. 4, November1986, 112 - 118

T. Dovan, T. S. Dillon, C. S. Berger, K. E. Forward, ‘A microcomputer based on-line identification approach to power system dynamic load modelling’, IEEETransactions on Power Systems, Vol. PWRS-2, No. 3, August 1987, 529 - 536

E. Vaahedi, M. A. El-Kady, J. A. Libaque-Esaine, V. F. Carvalho, ‘Load models forlarge-scale stability studies from end-user consumption’, IEEE Transactions onPower Systems, Vol. PWRS-2, No. 4, November 1987, 864 - 872

Juraj Medani´c, Marija Ili´c-Spong, John Christensen, ‘Dicrete models of slowvoltage dynamics for under load tap-changing transformer coordination’, IEEETransactions on Power Systems, Vol. PWRS-2, No. 4, November 1987, 873 - 882

Chu Liu, Qung-Min Zhang, ‘Two reduced-order methods for studies of powersystem dynamics’, IEEE Transactions on Power Systems, Vol. 3, No. 3, August1988, 894 - 899

Hill, D. J., ‘Nonlinear Dynamic Load Models with Recovery for Voltage StabilityStudies’, IEEE/PES 1992 Winter Meeting, New York, January 26-30, 1992

Appendix A

87

Appendix A

Street Lighting

Here is a compilation of a preliminary inventory of the street lighting in Göteborgmade by “Göteborg Energi AB”. Total rated load is 12.7 MW.

Table A.1 Incandescent lamp. Total rated power of incandescent lamps: 36 480 W.

Table A.2 Fluorescent lamp. Total rated power of fluorescent lamps: 77.3 kW.

Rated power [W] Number Total rated power [W]

60 23 1 380

75 34 2 550

100 81 8 100

150 23 3 450

500 20 10 000

1 000 11 11 000


20 41 820

30 16 480

32 27 864

40 1 542 61 680

65 207 13 455

Appendix A

88

Table A.3 Mercury lamp. Total rated power of mercury lamps: 9.3 MW.

Table A.4 Low pressure sodium lamp. Total rated power of low pressure sodium lamps:

1 076 kW.

Table A.5 High pressure sodium lamp. Total rated power of high pressure sodium lamps:

2 MW.


40 272 10 880

50 465 23 250

80 3 885 310 800

125 46 494 5 811 750

250 12 082 3 020 500

400 82 32 800

700 3 2 100

1000 76 76 000


55 656 36 080

90 3 958 356 220

91 5 477 498 407

135 1 000 135 000

180 281 50 580


50 158 7 900

70 2 226 155 820

100 596 59 600

150 3 057 458 550

215 82 17 630

250 3 013 753 250

400 1 369 547 600

Appendix A

89

Table A.6 Mixed colour lamp. Total rated power of mixed colour lamps: 180 380 W.

Table A.7 Metal halogen lamp. Total rated power of metal halogen lamps: 2 kW.


160 918 146 880

250 134 33 500


400 5 2 000

Appendix A

90

Appendix B

91

Appendix B

Inaccuracy due to measurement devices and measurement arrangement

B.1 The bias in voltage and current

The absolute error in the displayed values of the multimeters is dependent onmeasuring range and measured value. According to the manufacturer, the maximumlimit of the absolute error of the multimeters for sinewave inputs equals:

(300V-range)

(3A-range)

(300 mA-range)

where Um and Im are measured quantities.

[V]

[A]

B.2 The bias in active power

The absolute error in the displayed value of the wattmeter is dependent onmeasuring range and power factor, but not on measured value. According to themanufacturer, the maximum limit of the absolute error of the wattmeter is:

at cos ϕ = 1.0

Eu0.5100--------- Um⋅ 0.102+=

Ei1.5100--------- Im⋅ 0.00163+=

Ei0.81100---------- Um⋅ 0.000163+=

U Um Eu±=

I Im Ei±=

Ep1.0

2 0.1⋅100

--------------- AuAi⋅

2---------------------------------=

Appendix B

92

at cos ϕ = 0.7

at cos ϕ = 0.5

at cos ϕ = 0.1

where Au and Ai are the voltage and current range of the wattmeter respectively.

[W]

B.3 The bias in reactive power

The reactive power is calculated from measured values of voltage Um, current Imand active power Pm

To derive the maximum limit of the absolute error, the following formula was used

which results in

Ep0.7

2 0.15⋅100

------------------ AuAi⋅ 0.2+

2--------------------------------------------------=

Ep0.5

2 0.2⋅100

--------------- AuAi⋅ 0.2+

2-----------------------------------------------=

Ep0.1

2 0.5⋅100

--------------- AuAi⋅ 0.2+

2-----------------------------------------------=

P Pm Ep±=

Qc UmIm( )2Pm

2–=

EqdQdU-------

maxEu

dQdI-------

maxEi

dQdP-------

maxEp+ +=

Eq

IEu UEi

P Ep+( )Ep

U Eu–( ) I Ei–( )---------------------------------------- 2EiEu–+ +

1P Ep+( )2

U Eu–( )2I Ei–( )2

---------------------------------------------–

---------------------------------------------------------------------------------------------=

Q Qc Eq±=

Appendix B

93

B.4 The inaccuracy due to measurement arrangement

The inaccuracy due to the measurement arrangement is dependent on the resistancein the cords, the input resistance of the different measurement devices and thelocation of the measuring devices within the circuit. To obtain some idea of how themeasurement arrangement affects the relative error in the measured values, anaccuracy analysis was performed with a totally resistive load connected to thecircuit. The measurement devices were connected according to Figure B.1.

Figure B.1 The measurement arrangement.

The input resistance for the voltage and current input terminals of the multimetersare 1.0 MΩ and 0.3 Ω respectively, and the input resistance for the voltage andcurrent input terminals of the wattmeter are 660 kΩ and 0.01 Ω.

If the resistance in the cords are included, this will result in the equivalent circuitshown in Figure B.2.

Where

Ri ≈ 1.0 Ω

Ruw ≈ 660 kΩ // 660 kΩ coupled for double resolution

Riw ≈ 1 Ω

Ru ≈ 1.0 MΩ

G

W

VTestobject

A

Appendix B

94

Figure B.2 An equivalent circuit for the measurement arrangement.

For a purely resistive load, the quantities UL, IL and PL equal:

UL = Um

⇒

which give the relative error

These errors can be considered as negligible. No calculation has been carried out foran inductive/capacitive load.

Im IL

Ruw

Ri Riw

Ru

+

Um

–

+

UL

–

Resistiveload

Pm

PL

IL

Ruw

Ruw Riw+--------------------------

Im1

Ruw Riw+-------------------------- 1

Ru------+

Um– Im1

Ruw---------- 1

Ru------+

Um–≈=

PL ULIL

Ruw

Ruw Riw+--------------------------

ULIm1

Ruw Riw+-------------------------- 1

Ru------+

UL2

–= =

PL UmIm1

Ruw---------- 1

Ru------+

Um2

–≈

Im IL–

IL-----------------

4.03 106–Um⋅

Im 4.03 106–Um⋅–

-----------------------------------------------=

Pm PL–

PL--------------------

Pm UmIm– 4.03 106–Um

2⋅+

UmIm 4.03 106–Um

2⋅–----------------------------------------------------------------------=

Appendix C

95

Appendix C

Dynamic voltage tests

This appendix comprises figures that show the consumed active and reactive poweras a function of time when voltage steps of a magnitude of approximately –10% and+10% are applied to a test object.

Contents:

Figure C.1 Mercury lamp (400 W): voltage step of approx. –10%

Figure C.2 Mercury lamp (400 W): voltage step of approx. +15%

Figure C.3 High pressure sodium lamp (250 W): voltage step of approx. –15%

Figure C.4 High pressure sodium lamp (250 W): voltage step of approx. +10%

Figure C.5 Low pressure sodium lamp (91 W): voltage step of approx. –20%

Figure C.6 Low pressure sodium lamp (91 W): voltage step of approx. +20%

Figure C.7 Fluorescent lamp (2x36 W): voltage step of approx. –15%

Figure C.8 Fluorescent lamp (2x36 W): voltage step of approx. +10%

Figure C.9 Low-energy lamp (20 W): voltage step of approx. –10%

Figure C.10 Low-energy lamp (20 W): voltage step of approx. +15%

Figure C.11 Incandescent lamp (60 W): voltage step of approx. –10%

Figure C.12 Incandescent lamp (60 W): voltage step of approx. +10%

Figure C.13 Refrigerator: voltage step of approx. –20%

Figure C.14 Refrigerator: voltage step of approx. +20%

Appendix C

96

Figure C.1 Consumed active and reactive power as a function of time when a voltage step

of the magnitude of approx. –10% (230 V down to 210 V) is applied to a mercury lamp.


of the magnitude of approx. +15% (200 V up to 233 V) is applied to a mercury lamp.

Appendix C

97


of the magnitude of approx. –15% (230 V down to 200 V) is applied to a high pressure sodium

lamp.


of the magnitude of approx. +10% (205 V up to 230 V) is applied to a high pressure sodium

lamp.

Appendix C

98


of the magnitude of approx. –20% (230 V down to 190 V) is applied to a low pressure sodium

lamp.


of the magnitude of approx. +20% (190 V up to 225 V) is applied to a low pressure sodium

lamp.

Appendix C

99


of the magnitude of approx. –15% (230 V down to 200 V) is applied to a fluorescent lamp.


of the magnitude of approx. +10% (200 V up to 220 V) is applied to a fluorescent lamp.

Appendix C

100


of the magnitude of approx. –10% (230 V down to 208 V) is applied to a low-energy lamp.


of the magnitude of approx. +15% (208 V up to 235 V) is applied to a low-energy lamp.

Appendix C

101

Figure C.11 Consumed active power as a function of time when a voltage step of the

magnitude of approx. –10% (230 V down to 203 V) is applied to an incandescent lamp.

Figure C.12 Consumed active power as a function of time when a voltage step of the

magnitude of approx. +10% (203 V up to 228 V) is applied to an incandescent lamp.

Appendix C

102


of the magnitude of approx. –20% (230 V down to 183 V) is applied to a refrigerator.


of the magnitude of approx. +20% (183 V up to 227 V) is applied to a refrigerator.

Appendix D

103

Appendix D

Accuracy analysis

This appendix comprises figures that show the accuracy-variations in measuredactive power, reactive power and voltage. The crosses in the figures show maximumlimits of the absolute error for active power and voltage, and reactive power andvoltage, respectively, according to Figure D.1. The diagrams are presented in per-unit values. The base-values are P0, Q0 and U0, where P0 and Q0 aremeasured/calculated powers at nominal voltage U0 (~230 V).

Contents:

Figure D.1 The absolute error cross

Figure D.2 Mercury lamp (250 W): accuracy-variations in measured activepower and voltage

Figure D.3 Mercury lamp (250 W): accuracy-variations in measured reactivepower and voltage

Figure D.4 High pressure sodium lamp (400 W): accuracy-variations inmeasured active power and voltage

Figure D.5 High pressure sodium lamp (400 W): accuracy-variations inmeasured reactive power and voltage

Figure D.6 Low pressure sodium lamp (91 W): accuracy-variations in measuredactive power and voltage

Figure D.7 Low pressure sodium lamp (91 W): accuracy-variations in measuredreactive power and voltage

Figure D.8 Fluorescent lamp (2x36 W): accuracy-variations in measured activepower and voltage

Figure D.9 Fluorescent lamp (2x36 W): accuracy-variations in measuredreactive power and voltage

Appendix D

104

Figure D.10 Low-energy lamp (11 W): accuracy-variations in measured activepower and voltage

Figure D.11 Low-energy lamp (11 W): accuracy-variations in measured reactivepower and voltage

Figure D.12 Incandescent lamp (60 W): accuracy-variations in measured activepower and voltage

Figure D.13 Refrigerator: accuracy-variations in measured active power andvoltage

Figure D.14 Refrigerator: accuracy-variations in measured active power andvoltage

Figure D.1 The maximum limit of the absolute error, where Pm and Um are measured

active power and voltage, and Ep, Eu respectively are the calculated maximum limits of the

absolute error.

Figure D.2 The accuracy-variations in measured active power and voltage of a mercury

lamp. U0 = 230 V, P0 = 307 W.

Um–Eu Um+EuUm

Pm–Ep

Pm+Ep

Pm

Appendix D

105

Figure D.3 The accuracy-variations in measured reactive power and voltage of a mercury

lamp. U0 = 230 V, Q0 = 147 var.

Figure D.4 The accuracy-variations in measured active power and voltage of a high

pressure sodium lamp. U0 = 230 V, P0 = 510 W.

Appendix D

106

Figure D.5 The accuracy-variations in measured reactive power and voltage of a high

pressure sodium lamp. U0 = 230 V, Q0 = 170 var.

Figure D.6 The accuracy-variations in measured active power and voltage of a low

pressure sodium lamp. U0 = 230 V, P0 = 121 W.

Appendix D

107

Figure D.7 The accuracy-variations in measured reactive power and voltage of a low

pressure lamp. U0 = 230 V, Q0 = 60 var.

Figure D.8 The accuracy-variations in measured active power and voltage of a fluorescent

lamp. U0 = 230 V, P0 = 100 W.

Appendix D

108

Figure D.9 The accuracy-variations in measured reactive power and voltage of a

fluorescent lamp. U0 = 230 V, Q0 = 55 var.

Figure D.10 The accuracy-variations in measured active power and voltage of a low-energy

lamp. U0 = 230 V, P0 = 11 W.

Appendix D

109

Figure D.11 The accuracy-variations in measured reactive power and voltage of a low-

energy lamp. U0 = 230 V, Q0 = 22 var.

Figure D.12 The accuracy-variations in measured active power and voltage of incandescent

lamp. U0 = 230 V, P0 = 60 W.

Appendix D

110

Figure D.13 The accuracy-variations in measured active power and voltage of a refrigerator

lamp. U0 = 230 V, P0 = 85 W.

Figure D.14 The accuracy-variations in measured reactive power and voltage of

refrigerator. U0 = 230 V, Q0 = 148 var.

Appendix E

111

Appendix E

Wave forms

This appendix comprises figures that show the wave forms of applied voltage andcurrent to a test object.

Contents:

Figure E.1 Mercury lamp

Figure E.2 High pressure sodium lamp

Figure E.3 Low pressure sodium lamp

Figure E.4 Fluorescent lamp

Figure E.5 Low-energy lamp

Figure E.6 Refrigerator

Appendix E

112

Figure E.1 The wave form of the applied voltage and current to a mercury lamp.

Figure E.2 The wave form of the applied voltage and current to a high pressure sodium

lamp.

Ampl

itude

Time

Ampl

itude

Time

Appendix E

113

Figure E.3 The wave form of the applied voltage and current to a low pressure sodium

lamp.

Figure E.4 The wave form of the applied voltage and current to a fluorescent lamp.

Ampl

itude

Time

Ampl

itude

Time

Appendix E

114

Figure E.5 The wave form of the applied voltage and current to a low-energy lamp.

Figure E.6 The wave form of the applied voltage and current to a refrigerator.

Ampl

itude

Time

Ampl

itude

Time

Appendix F

115

Appendix F

Derived static load models

In this appendix the results from the static measurements are presented with thederived static load models.

Each figure shows the results from one static measurement and one derived loadmodel. For mercury lamps and refrigerators all models in Section 3.6.5 arepresented, (exponential, linear, and polynomial load model) while for the otherlamps only the exponential load model is presented. Note that for high- and lowpressure sodium lamps the measurements for reactive power are not presented dueto the poor accuracy.

In the diagrams the datums are presented with a star “*” if it was taken into accountwhen deriving the model, if not it is presented with a cross “+”. The diagrams arepresented in per-unit values. The base-values are P0, Q0 and U0, where P0 and Q0are measured/calculated powers at nominal voltage U0 (~230 V).

Contents:

Figure F.1 Mercury lamp (250 W): active power and exponential load model

Figure F.2 Mercury lamp (250 W): active power and linear load model

Figure F.3 Mercury lamp (250 W): active power and polynomial load model

Figure F.4 Mercury lamp (250 W): reactive power and exponential load model

Figure F.5 Mercury lamp (250 W): reactive power and linear load model

Figure F.6 Mercury lamp (250 W): reactive power and polynomial load model

Figure F.7 High pressure sodium lamp (400 W): active power and exponentialload model

Appendix F

116

Figure F.8 Low pressure sodium lamp (91 W): active power and exponentialload model

Figure F.9 Fluorescent lamp (2x36 W): active power and exponential loadmodel

Figure F.10 Fluorescent lamp (2x36 W): reactive power and exponential loadmodel

Figure F.11 Low-energy lamp (11 W): active power and exponential load model

Figure F.12 Low-energy lamp (11 W): reactive power and exponential loadmodel

Figure F.13 Incandescent lamp (60 W): active power and exponential load model

Figure F.14 Refrigerator, active power and exponential load model

Figure F.15 Refrigerator, active power and linear load model

Figure F.16 Refrigerator lamp: active power and polynomial load model

Figure F.17 Refrigerator: reactive power and exponential load model

Figure F.18 Refrigerator: reactive power and linear load model

Figure F.19 Refrigerator, reactive power and polynomial load model

Appendix F

117

Figure F.1 Recorded active power for a mercury lamp and the derived exponential model,

np = 2.297. Least square error = 5.9 · 10–3, U0 = 230 V, P0 = 307 W.

Figure F.2 Recorded active power for a mercury lamp and the derived linear model,

a0 = –1.327, a1 = 2.327. Least square error = 1.6 · 10–2, U0 = 230 V, P0 = 307 W.

Appendix F

118

Figure F.3 Recorded active power for a mercury lamp and the derived polynomial model,

a0 = 0.1260, a1 = –0.5480, a2 = 1.422. Least square error = 5.9 · 10–3, U0 = 230 V,

P0 = 307 W.

Figure F.4 Recorded reactive power for a mercury lamp and the derived exponential

model, nq = 5.577. Least square error = 4.1 · 10–3, U0 = 230 V, Q0 = 147 var.

Appendix F

119

Figure F.5 Recorded reactive power for a mercury lamp and the derived linear model, b0 =

–4.015, b1 = 5.915. Least square error = 1.1 · 10–1, U0 = 230 V, Q0 = 147 var.

Figure F.6 Recorded reactive power for a mercury lamp and the derived polynomial

model, b0 = 5.470, b1 = –14.59, b2 = 10.12. Least square error = 2.2 · 10–2, U0 = 230 V,

Q0 = 147 var.

Appendix F

120

Figure F.7 Recorded active power for a high pressure sodium lamp and the derived

exponential model, np = 2.262. Least square error = 1.5 · 10–2, U0 = 230 V, P0 = 510 W.

Figure F.8 Recorded active power for a low pressure sodium lamp and the derived

exponential model, np = 0.41. Least square error = 1.5 · 10–2, U0 = 230 V, P0 = 121 W.

Appendix F

121

Figure F.9 Recorded active power for a fluorescent lamp and the derived exponential

model, np = 2.034. Least square error = 8.7 · 10–3, U0 = 230 V, P0 = 100 W.

Figure F.10 Recorded reactive power for a fluorescent lamp and the derived exponential


Appendix F

122

Figure F.11 Recorded active power for a low-energy lamp and the derived exponential


Figure F.12 Recorded reactive power for a fluorescent lamp and the derived exponential


Appendix F

123

Figure F.13 Recorded active power for an incandescent lamp and the derived exponential


Figure F.14 Recorded active power for a refrigerator and the derived exponential model,

np = 1.583. Least square error = 4.8 · 10–2, U0 = 230 V, P0 = 85 W.

Appendix F

124

Figure F.15 Recorded active power for a refrigerator and the derived linear model,

a0 = –0.62, a1 = 1.62. Least square error = 5.3 · 10–2, U0 = 230 V, P0 = 85 W.

Figure F.16 Recorded active power for a refrigerator and the derived polynomial model,

a0 = 2.190, a1 = –3.955, a2 = 2.765. Least square error = 2.8 · 10–2, U0 = 230 V, P0 = 85 W.

Appendix F

125

Figure F.17 Recorded reactive power for a refrigerator and the derived exponential model,

nq = 3.255. Least square error = 3.6 · 10–2, U0 = 230 V, Q0 = 148 var.

Figure F.18 Recorded reactive power for a refrigerator and the derived linear model,

b0 = –2.370, b1 = 3.370. Least square error = 8.2 · 10–2, U0 = 230 V, Q0 = 148 var.

Appendix F

126

Figure F.19 Recorded reactive power for a refrigerator and the derived polynomial model,

b0 = 2.190, b1 = –5.650, b2 = 4.460. Least square error = 4.2 · 10–2, U0 = 230 V, Q0 = 148

var.

Appendix G

127

Appendix G

Evaluation of the equations for up-curves and qu-curves

Assume the system of one power line represented according to Figure G.1.

Figure G.1 A single line system.

P and Q are the transmitted power on the line, expressed by the following equations

(G.1)

(G.2)

where the loss-angle

Solving sin2(ψ + δ) + cos2(ψ + δ) = 1 gives

(G.3)

E U

P / Q

Z = R +jXL L Lψ

3ψ

2

PU

2

ZL-------sinδ–

EUZL--------sin ψ δ+( )+=

QU

2

ZL-------cosδ–

EUZL--------cos ψ δ+( )+=

δ arctanRL

XL-------=

P2

Q2

+ E2U

2

ZL2

------------- U4

ZL2

-------– 2U

2

ZL------- Psinδ Qcosδ+( )–=

Appendix G

128

with E as base-voltage and ZL as base impedance giving E2/ZL as base-power,Equation (G.3) with per-unit values is

p2 + q2 = u2 [1 – u2 – 2(p sinδ + q cosδ)] (G.4)

rearranged

p2 = u2 [1 – u2 – 2(p sinδ + q cosδ)] – q2 (G.5)

If the line is considered lossless, i.e. R = 0 giving δ = 0, Equation (G.5) turns to

p2 = u2 – (u2 + q)2 (G.6)

G.1 UP-curve

Solve Equation (G.3) for U

(G.7)

where

with per-unit values

(G.8)

and for the lossless line

(G.9)

In Equations (G.7), (G.8) and (G.9) the reactive power can be expressed as aconstant power or as a function of p, q = f(p), e.g. q = p · tanϕ where tanϕ is

UE

22ZLA–

2---------------------------

E2

2ZLA+( )2

4----------------------------------- ZL

2P

2Q

2+( )–±=

A Psinδ Qcosδ+=

u 12--- A– 1

4--- A

2A– p

2q

2+( )–+±=

A psinδ qcosδ+=

u 12--- q– 1

4--- q– p

2–±=

Appendix G

129

constant. With q = p · tanϕ and the possibility of changing the feeding voltage E toa new value Enew Equations (G.8) and (G.9) become

(G.10)

where

and

(G.11)

To find the curve connecting the points with pmax, return to Equation (G.4), but addthe possibilities of changing E and q = f(p), i.e.

(G.12)

by implicit differentiation

(G.13)

At pmax dp/du = 0 and therefore

(G.14)

and

(G.15)

uenew

2

2----------- pA–

enew4

4----------- p

2A

2enew

2pA– p

21 tan

2ϕ+( )–+±=

A sinδ tanϕcosδ+=

uenew

2

2----------- ptanϕ–

enew4

4----------- enew

2ptanϕ– p

2–±=

p2

f2

p( )+ u2

enew2

u2

– 2 psinδ f p( )cosδ+[ ]–

=

dpdu------

2uenew2

4u3

– 4upsinδ– 4uf p( )cosδ–

2p 2f p( )df p( )dp

------------- 2u2sinδ 2u

2df p( )dp

-------------cosδ+ + +------------------------------------------------------------------------------------------------------------=

2u enew2

2u2

– 2pmaxsinδ– 2f p( )cosδ–( ) 0=

f p( )enew

22u

2– 2pmaxsinδ–

2cosδ----------------------------------------------------------=

Appendix G

130

Inserted in Equation (G.12) this gives

(G.16)

(G.17)

For the lossless line, R = 0, Equation (G.17) is reduced to

(G.18)

and

(G.19)

or if Enew = E

(G.20)

When R ≠ 0, solve Equation (G.17) for u

(G.21)

where

(G.22)

where

pmax2 enew

22u

2– 2pmaxsinδ–( )

2

4cos2δ

-----------------------------------------------------------------+ u4

=

u4

1 cos2δ–( ) u

2enew

22pmaxsinδ–( )–

enew4

4----------- enew

2pmaxsinδ+–+

+pmax2

sin2δ cos

2δ+( ) 0=

u2enew

2–

enew4

4----------- pmax

2+ + 0=

up pmax=

pmax2

enew2

-----------enew

2

4-----------+=

up pmax= pmax2 1

4---+=

up pmax=

enew2

2A–

2 1 cos2δ–( )

-------------------------------enew

22A–( )

2

4 1 cos2δ–( )

2----------------------------------

enew2

Aenew

4

4-----------– pmax

2–

1 cos2δ–

---------------------------------------------------+±=

A pmaxsinδ=

up pmax=

enew2

2A–

2sin2δ

------------------------enew

2

sinδ-----------

enew2

4sin2δ

----------------pmax

sinδ-----------– A

enew2

4-----------–+±=

A pmaxsinδ=

Appendix G

131

or if Enew = E

(G.23)

If q or q = f(p) is known, pmax can be expressed. Assume q = p · tanϕ and isolate uin Equation (G.14)

(G.24)

Insert Equation (G.24) in Equation (G.12) together with f(p) = p · tanϕ

where

(G.25)

(G.26)

(G.27)

up pmax=

1 2pmaxsinδ–

2sin2δ

----------------------------------1

sinδ---------- 1

4sin2δ

----------------pmax

sinδ-----------– pmaxsinδ 1

4---–+±=

u2 enew

2

2----------- pmax sinδ tanϕcosδ+( )–=

pmax2

1 tan2ϕ+( )

enew2

2----------- A–

enew2 enew

2

2----------- A–

2

– 2enew

2

2----------- A–

A =–=

= e new4

4

----------- e new2

A– A 2

+ for a positive power according to referencesin=

Chapter 5e

new2

2

----------- A–

2

p

max2

cos

2

ϕ

--------------= =

A pmax sinδ tanϕcosδ+( )=

pmax

cosϕ-----------

enew2

2----------- pmax sinδ tanϕcosδ+( )–=

enew2

2----------- pmax sinδ tanϕcosδ 1

cosϕ-----------+ +

=

pmax

enew2

cosϕ2 cosϕsinδ sinϕcosδ 1+ +( )---------------------------------------------------------------------

enew2

cosϕ2 sin ϕ δ+( ) 1+[ ]------------------------------------------= =

Appendix G

132

G.2 QU-curves

Solve Equation (G.3) for Q

(G.28)

with per-unit values

(G.29)

and for a lossless line

(G.30)

Henceforth only the lossless line will be dealt with concerning qu-curves.

In Equation (G.30) the active power can be expressed as a constant power or as afunction of the reactive power, p = f(q), e.g. p = q/tan

ϕ

where tan

ϕ

is constant.

With p = q/tan

ϕ

and the possibility of changing the feeding voltage E to E

new

Equation (G.30) becomes

(G.31)

Q U2cosδ

ZL------------------–

U4

ZL2

------- cos2δ 1–( ) U

2

ZL------- E

2

ZL------ 2Psinδ–

P2

–+±=

q u2cosδ– u

4cos

2δ 1–( ) u2

1 2psinδ–( ) p2

–+±=

q u2

– u2

p2

–±=

q u2

1 tan2ϕ+

-----------------------– u4

1 tan2ϕ+( )

2-------------------------------

u2enew

2u

4–

1 tan2ϕ+

-----------------------------+±=

Appendix H

133

Appendix H

Evaluation of the equations for the impact of tap changer on system

characteristics

Recall Figure 5.17 in Chapter 5

Figure H.1 A system with an equivalent voltage Ei′ behind a reactance Zi′, both reduced

over the nominal ratio of the transformer, and a tap changer regulating the voltage E.

Ei′ and Zi′ are reduced to the voltage level of the single line to the right of bus two.To write the power flow equations, reduce the voltage Ei′ and the reactance Zi′ overthe tap changer ratio according to Figure H.2.

Figure H.2 The system in Figure H.1 but with Ei′ and Zi′ reduced over the tap changer

ratio.

Z′i(1+∆i):1

P/QZL

1 2 3

E′i E U

(1+∆) E LZ U

P / Q

iZ'iE'

1 2 3

(1+∆i)2

Appendix H

134

and the power flow equations will be

(H.1)

(H.2)

where

(H.3)

and δ is the loss-angle for the total system.

Solving sin2(ψ + δ) + cos2(ψ + δ) = 1 and writing the equation with per unit values(Ei′ as base-voltage and ZL as base-reactance giving (Ei′)2/ZL as base-power) andwith the possibility of changing the feeding voltage Ei′ to a new value Ei′(new) gives

(H.4)

where is not constant when ∆i varies.

Isolate u to get the equation for the up-curve

(H.5)

where

For the sake of simplicity the resistance is neglected and the impedance Xi′ iswritten as αXL giving

(H.6)

PU

2

Z-------sinδ–

E′iU1 ∆i+( )Z

-----------------------sin ψ δ+( )+=

QU

2

Z-------cosδ–

E′iU1 ∆i+( )Z

-----------------------cos ψ δ+( )+=

ZZ′i

1 ∆i+( )2---------------------- ZL+=

p2

q2

+

e′i(new)( )2 1

1 ∆i+( )2----------------------u

2u

4–

z2

------------------------------------------------------------------ 2u psinδ qcosδ+( )z

----------------------------------------------–=

z ZZL------=

u

e′i(new)( )2

1 ∆i+( )2------------------------- 2zA–

2------------------------------------------

e′i(new)( )2

1 ∆i+( )2------------------------- 2zA–

2

4-------------------------------------------------- z

2p

2q

2+( )–±=

A psinδ pcosδ+=

X XLα

1 ∆i+( )2---------------------- 1+=

Appendix H

135

together with the assumption that q = p · tanϕ, and Equation (H.5) becomes

(H.7)

where

To find the expression for pmax, recall Equation (H.4) and add q = p · tanϕ. Byimplicit differentiation

(H.8)

at pmax dp/du = 0 and therefore

(H.9)

Isolate u2/z

(H.10)

Insert Equation (H.10) in Equation (H.4) with q = p · tanϕ

⇒

u

1

ni2

----- 2 α

ni2

----- 1+

ptanϕ–

2--------------------------------------------------- 1

4ni4

---------1

ni2

----- α

ni2

----- 1+

ptanϕ– α

ni2

----- 1+ 2

p2

–±=

ni2

1 ∆i+( )2=

dpdu------

2uz

------

e′i(new)( )2 1

1 ∆i+( )2---------------------- 2u

2–

z--------------------------------------------------------------- 2p sinδ tanϕcosδ+( )–

2p 1 tan2δ+( ) 2u

2

z--------- sinδ tanϕcosδ+( )+

-------------------------------------------------------------------------------------------------------------------------------------=

e′i(new)( )2 1

1 ∆i+( )2---------------------- 2u

2–

z--------------------------------------------------------------- 2pmax sinδ tanϕcosδ+( )– 0=

u2

z-----

12z------ e′i(new)( )2 1

1 ∆i+( )2---------------------- pmax sinδ tanϕcosδ+( )–=

pmax2

1 tan2ϕ+( )

pmax2

cos2ϕ

-------------- ==

= 1

4z

2 --------

e

′

i(new) ( )

4

1

∆

i

+

( )

4

------------------------- 1z---

e

′

i(new) ( )

2

1

∆

i

+

( )

2

-------------------------p max A– p max2

A 2

+

Appendix H

136

(H.11)

Where

(H.12)

(H.13)

If the resistance is neglected and the total impedance is written according toEquation (H.6) then p

max

can be written as

(H.14)

pmax2

cos2ϕ

--------------1

2z------

e′i(new)( )2

1 ∆i+( )2------------------------- pmaxA–

2

=

A sinδ tanϕcosδ+=

pmax

cosϕ-----------

12z------ e′i(new)( )2 1

1 ∆i+( )2---------------------- pmax sinδ tanϕcosδ+( )–=

pmax

e′i(new)

1 ∆i+-----------------

2

1z--- cosϕ

2 sin ϕ δ+( ) 1+[ ]------------------------------------------=

pmax e′i(new)( )2 1α 1 ∆i+( )+----------------------------- cosϕ

2 sinϕ 1+( )----------------------------=

Appendix I

137

Appendix I

Exponential load model

In Chapter 5 an exponential load model is used. The load characteristics for activeand reactive power are then

(I.1)

(I.2)

where n

p

and n

q

are the sensibility to voltage, and u is the voltage feeding the load.For nominal voltage u

0

, the rated power p

0

and q

0

are given respectively.

If n

p

and n

q

are not equal the relationship between p and q may be derived oncondition that p0 and q0 are known, p/p0 > 0 and q/q0 > 0, and given at the samenominal voltage u0. Then

(I.3)

and

(I.4)

(I.5)

p p0uu0-----

np

=

q q0uu0-----

nq

=

uu0-----

pp0-----

1np-----

qq0-----

1nq-----

= =

p p0qq0-----

np

nq-----

=

q q0pp0-----

nq

np-----

=

Appendix I

138

Documents

Modelling of Load Devices and Studying Load/System Characteristics