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Investigation and Control of the Damping of Power System Oscillations
IF AC Symposium 1977 Melbourne, 21-25 February 1977
Investigation and Control of the Damping of Power System Oscillations A .T . WILSON, G.D . WHITE, G.M. BREUER and R.L. BOLDEN State Electricity Commission of Victoria, Melbourne
SUMMARY The development of a design approach to the investigation and control of the damping of power system oscillations is given, including an outline of the analysis technique used and the approach followed in setting limits to system operation. age through control of field voltage, effectively reduce the equivalent damping contribution provided to the system by the machine damper windings for some of these oscillation modes.
INTRODUCTION Limits to the operational flexibility of power systems have classically been defined in terms of power transfer capability between generation and load centres, or between regions of a large interconnected system. In closely coupled systems the limits have been determined by contingent thermal loading of the transmission. However, in weakly coupled systems, the limits have been determined by system phenomena generally categorised in terms of steady state and transient stability of the generators, loads and the transmission network following the loss of the most critical element.
Under certain system conditions these oscillation modes may be excited by the small perturbations that are continually occurring in the system. If the damping associated with a particular oscillation mode is low, the system will oscillate for prolonged periods. Systems characterised by large generation and load centres located electrically at remote ends of a relatively weak transmission system can be particularly prone to this phenomena when transferring power between the centres. Recent examples of this have been experienced in the interconnected systems of Italy and Yugoslavia (ref. 2), and the New South Wales/Snowy/Victoria interconnected systems in A~tralia (ref. 3).
The classical definition of steady state limits have been understood in terms of the 90 degree limit of rotor angle between equivalent generator rotors at either end of the transmission network being studied. In large, multi-machine interconnected systems, however, constraints arising from limited dynamic reactive reserve capability have resulted in a need to re-assess the limits in terms of contingent voltage profile and reactive power losses in the transmission following loss of the most critical element in the network, particularly at periods of high demand in the system.
y
j'
Ensuring ~. adequate understanding of this phenomena and its effect on limits of system operation has required the development, testing, application and interpretation of digital computer based analytical tools. 2
When faults resulting in loss of a major transmission element occur, the fault severity, its duration and the resultant impedance change in the network all contribute to the transient stability performance of the system. Duplicated protection and fast acting circuit breakers have reduced significantly the effect of the fault itself, and the recent development of fast acting excitation systems has further enhanced the overall transient stability performance of the system.
ANALYSIS TECHNIQUES
The normal approach to the study of oscillation damping problems is to use an equivalent linearised system rather than directly solve the basic nonlinear differential equations describing the system. This can then be analysed by classical control system methods, i.e. Nyquist Criterion, Routh Criterion, Root Locus, Domain Separation or Eigenval ue Analysis. The amount of computation required for each method is considerable and it has been common practice to further simplify the analysis by reducing the system to a single machine connected via an impedance to an infinite bus. This simplification is valuable for detailed analysis of an individual machine and regulator, but may not necessarily give an accurate indication of behaviour on a system basis where interactions between machines or groups of machines can occur. It is in this area of system behaviour that a more complete multi-machine analysis is required. For such studies eigenvalue analysis which uses a state space matrix formulation of the equations has computational advantages over the other methods. The use of matrix formulation facilitates the handling of the large number of variables involved and allows the overall system
The need to ensure well-damped and stable operation of the excitation system and synchronous machine When connected to the system has necessitated either a trade off in terms of excitation system response through internal voltage regulator feedback loops or the application of special damping control signals (ref. 1). Further, the power system has a large number of possible natural modes of oscillation associated with the electro-mechanical characteristics of each generator, its associated controls and the electrical parameters of the network connections between them. High gain excitation systems, when connected as a control device regulating generator terminal volt-
49
within the program, they take the form of a series of submatrices down the diagonal with cross coupling terms given by the current equations.
matrix to be built up from sub-matrices describing each section. 2.1
Program Details 2.4
The initial analysis of the oscillation damping problem utilised a small scale multi-machine computer model developed in 1970 as part of a research project (ref. 4). This proved a valuable tool and allowed a good understanding of the phenomena involved, but was somewhat limited due to restriction on size and the relatively simple machine and AVR modelling used. Consequently, a more advanced and sophisticated program has been developed using the same basic approach but with increased size, more flexible modelling and improved input-output features. Details of the main aspects of the program are given below. 2.2
AVR information is expressed in terms of a block diagram. Two block types are recognised as shown in Fig. 1.
A.v'=:0-
Q . - TRANSFER FUNCTlON BLOCK
b .- SUMMER BLOCK
Data Input Fig. 1
Machine Equations
The program processes the AVR data block by block. For each input and output the variables already in the matrix are scanned to locate the variable required. If it is not found the next unused row and column in the matrix is assigned to the variable and the appropriate equation is entered.
MACHINE MODELS
Description
Standard AVR Blocks
By setting one or more of the constants A, B, C, D to zero, the transfer function block can be made to represent a gain, lag block, integrator, etc.
TABLE I
After all the blocks have been processed for one particular AVR, the unwanted (non state variable) variables, including those for the related machine, are eliminated and the matrix compressed.
Flux Variables Included
2.5 1
Vo
C .Ut
The machine representation is based on standard two axis representation (ref. 5). Six machine models of differing complexity and accuracy are available . to allow flexibility of studies (see Table I).
Model
L
II.V1.
A load flow program is used to set up the required operating point and the reduced admittance matrix is produced from a transient stability program. 2.3
AVR Equations
Calculation of Eigenvalues and Eigenvectors
-
Constant voltage behind reactance
2
Id-axis wdg.
1jJ'
3
Id-axis, Iq-axis wdg.
1jJ'd' 1jJ" q
4
2d-axis, Iq-axis wdg.
1jJ'd' 1jJ" d' 1jJ" q
Eigenvalues of the total system matrix are calculated using a standard QR algorithm. Only eigenvalues which are poorly damped are displayed. As an aid to interpretation of result the optional calculation of eigenvectors within a specified frequency band is available.
5
2d-axis, 2q-axis wdg.
1jJ'd' 1jJ" d' 1jJ' q' 1jJ" q
6
Infinite bus representation
3
d
-
VALIDATION OF RESULTS
The following methods of validation are available: 3.1
In the machine equations the effects of stator p1jJ terms and saturation have been neglected because their effect is small for dynamic phenomena. The effect of speed terms in the mechanical and voltage equations have been incorporated.
Comparison with transient (Time-Domain) Program
To properly test the eigenvalue program a full transient stability program including the effect of AVRs must be used; however, the problems associated with separating out the various oscillation modes and calculating their relative damping makes its use of doubtful value.
The effect of governors has also been neglected as their effect on oscillations of the frequencies being considered (0.2 - 2.0 Hz) is negligible. Mechanical damping due to the turbine is allowed for by a speed proportional term in the swing equation. In most studies this speed proportional damping terc is not increased to represent subtransient electrical damping effects because this practice tends to mask the oscillatory phenomena associated with AVR/machine/system interactions.
3.2
Comparison with Single Machine Infinite Bus Model
Various single machine - infinite bus models are available which have been exhaustively tested against each other and against practical tests. Comparison with these programs is valuable for verifying the machine and AVR modelling but does not enable checking of the intermachine behaviour nor an assessment of the effects of assumptions made in setting up the model (e.g. the effects of loads) .
For system studies the machine models used are 2 and 4. Model 2 gives a transient representation, while Model 4 is the normal sub transient representation which includes the effect of damper windings. Neglecting the effect of damper windings gives poor damping on machine local oscillation modes, particularly on synchronous condensers.
3.3
As these equations are placed in the system matrix
Comparison with Actual Power System Performance
If incidents of poor damping oscillations have
50
4.3
occurred, then these can be used as test cases, providing adequate information about the system operating point can be assembled. Although system tests are often difficult to achieve because of the problems of control and instrumentation, they are an essential and integral requirement of program validation.
In this mode, generation areas swing against other generation areas in a loosely coupled mode. An example of this oscillation mode would be Victorian generation swinging as a group against New South Wales generation via the interconnection.
Even with the data from an incident or system test, the eigenva1ue program will normally indicate lower levels of damping than those actually observed. This is due to sources of damping such as machine rotor body effects and the contribution of loads not being explicitly represented in the program. Rather than attempt to represent these effects by the usual speed proportional damping term, results from known system conditions are used to define reference damping levels, which then give the relative increase or decrease in damping for other system conditions. 4
Multi-Machine System Modes
It is in the analysis of multi-machine system modes that the eigenvalue program is most valuable Multi-machine local modes, although of interest, are generally well damped. Single machine local modes can be more conveniently studied on a single machine - infinite bus basis. 5
PRINCIPLES OF INVESTIGATIONS
5.1
Evaluation of Transfer Limits
Oscillation damping is sensitive to power transfers between regions, and it can be the determining phenomena for the limit of transfer. Before an assessment of the transfer limits on the systeu can be undertaken, it is first necessary to identify the system conditions at which poor damping of oscillations is likely to occur. Typically, these conditions are characterised by machines operating heavily loaded and close to their underexcited limits with the system at light load and heavy transfers occurring over long distances or over weak links.
INTERPRETATION OF RESULTS
Interpretation of results from the multi-machine eigenvalue program becomes difficult for high order systems, where up to 20 machines may be represented. As such systems can yie1d,over 100 eigenvalues, it becomes difficult to interpret which particular system components influence particular eigenva1ues. Although the problem is alleviated to some extent by the re~lisation that the system behaviour is primarily governed by lightly damped eigenvalues which are close to the imaginary axis in the eigenvalue plane, there may be several such eigenva1ues, which makes interpretation difficult.
Because of the sensitivity of damping to system operating conditions, it is essential to set up load flows that accurately represent a realistic system. Operating conditions are then studied at various power transfer levels for worst system contingencies, which may be loss of a generator it the importing system leading to an increase in transfer, or loss of a major system interconnecto] A useful analysis aid is to plot system damping (i.e. damping ratio of dominant system eigenvalue) as a ~nction of power transfer, as shown in Figure 3lor the critical contingency.
The problem of interpreting results is made much easier by the calculation of eigenvectors for each of the lightly damped eigenva1ues. This enables the influence of system components on particular eigenvalues to be better understood. In particular, it enables oscillation modes for each machine on the system to be determined. From system studies using this technique, it has been possible to divide machine oscillation modes into three broad categories as shown in figure 2. The influence of a particular machine on a mode is indicated by the magnitude and phase of its eigenvector relative to the eigenvectors of other machines.
OAhIf'ING OF DOMINANT OSCILLATION MODE
+
\'
--- ---
..~ (;l SIHCLE MACltIl4E L.OCAL OSCILLATION IoIOOE
(iil IlULTI-MACHINE LOCAL OSCILLATION MODE
~------
llPOIrr TO
EXPORT
GlON
- ~'\
(m) MULTI- MACHINE SYSTEM OSCILLATION MODE TRANSFER UMrrs
Fig. 2 4.1
Speed Eigenvectors for Specific Oscillation Modes
Fig. 3
~
+
TIE - WriE POWER FLOW .
System Damping as a Function of Interchange Between Regions
Single Machine Local Modes Setting up a system study for analysis will usual ] ~equire representation of at least one equivalent machine at each of the major generating stations and representation of equivalent synchronous compensators where these are judged important.
In this mode a single machine oscillates against the system with all other machines contributing very little. This is ana1agous to a single machine oscillating on an infinite bus. This mode tends to exist only for small machines and machines that are remotely situated from other machines. 4.2
FROM REGION
~
It is necessary to determine the sensitivity of system oscillatory damping to various system conditions. This sensitivity can then be used to determine constraints that need to be placed on certain system parameters for particular transfer conditions.
Multi-Machine Local Modes
This mode involves one machine group swinging against another machine group where the two machine groups are tightly coupled.
51
5.1.1
5.3
Changes in network structure
The system analysis studies that have been carried out indicate that the accuracy of machine and excitation control system data is especially important for the larger machines on the system because of their dominant contribution to system oscillation modes.
Outages of critical interconnectors have a detrimental effect on system damping to the extent that such outages may significantly reduce transfer capability. Outages of non-critical lines do not have such an effect and in fact may increase system damping at light load if the removal of the additional line charging from the system results in some machines operating in a more stable region. 5.1. 2
6
Possible stabilisation schemes to improve system damping are evaluated. The most cost effective scheme at present is the installation of power stabilisers on appropriate machines on the system. A detailed design of the stabilisation scheme is then undertaken. For the installation of a power stabiliser, this would involve a detailed redesign of the excitation control system on a single machine basis.
Generation distribution
Base load plant remote from load centres is inherently less stable than equivalent generation situated electrically close to the major load centres. This can necessitate the reduction of transfer limits for undesirable generation distributions or the restriction of station output on remote generation for a given transfer limit. 5.1. 4
IMPROVEMENT OF SYSTEM DAMPING
A strategy used for the improvement of system damping is detailed below:
Network loading and distribution
System loading level and load distribution affect the loading of generators and lines. At light system loads, where lines may be well below their surge impedance loading, generators may be required to operate in an under-excited mode to help maintain system voltages at acceptable levels. It is in this operating mode that system damping can become unacceptably low. 5.1. 3
The effectiveness of the stabilisation scheme is then examined using the multi-machine eigenvalue program, for all system operating conditions. The stabilisation scheme is then placed into service and its effectiveness assessed in practice by system tests.
Voltage levels
It has been found that for the region exporting power across an interconnection, the raising of voltage levels at base load power stations is in general stabilising. The effect of changing voltage on the receiving end of the system is not, however, so well defined. 5.1. 5
The multi-machine eigenvalue program is again used to reassess system operating limits with the modified excitation control system in service.
Distribution of inertia 7
System damping is also influenced by system inertia and its distribution. For example, the dominant system oscillation modes will be different for a system where all machines are operating close to full output than for a system of the same total generation with a larger number of machines operating with some machines at reduced output, i.e. low system nett inertia per MW.
7.1
DESIGN OF SYSTEM STABILISING DEVICES Choice of Stabilising Signal
System damping may be enhanced by introducing stabilising signals into the excitation systems of generators. The basic concept is to vary the terminal voltage of generators in phase with speed deviations and so provide damping action.
All the above aspects must be considered when evaluating transfer limits on the system if the advantages of interconnection are to be fully exploited. 5.2
Accuracy of Data
There are three quantities available local to a generator which, in conjunction with suitable signal processing, may be used for stabilising signals. These are generator rotor speed, electrical frequency at the generator terminals and electrical power output. Since, for a given level of system oscillation, the relative deviation in electrical power output is much greater than the relative speed and frequency changes, electrical power is the signal which should present the least problems with quiescent noise from transducers and rotor torsional oscillations. Instantaneous power transducers are readily available and can be installed in existing current and voltage transformer circuits, giving the power signal a distinct advantage over the speed signal, which requires custom built measuring equipment attached to the turbo-generator shaft. Of the three signals available, electrical frequency is the least sensitive to system oscillations and it is difficult to obtain a noise-free frequency signal without introducing significant delays due to filtering circuits. For these reasons, our attention has focussed primarily on stabilising signals derived from electrical power output for introduction into existing excitation systems.
Studying of Incidents
The following procedure has been adopted for analysing oscillation events: Data is collected on system operation conditions just prior to the onset of oscillations. The oscillation event is then simulated using the multi-machine eigenvalue program and the damping of the simulated event is compared to the actual event. The major factors contributing to the poor damping are analysed and interim operating restrictions are defined to ensure that the si tuation does not occur again. If necessary, a follow-up strategy is employed to improve the system damping and extend the allowable range of system operation (see Section 6).
52
7.2
as a function of frequency is given in Figure 4. It can be seen that the stabilised regulator has greatly increased the damping torque in the critical oscillation frequency range.
Stabilisation Using Power Signals
Because of the relatively long time constant of the field of a turbo-generator, the field voltage should be varied in phase with acceleration so that the terminal voltage varies in phase with speed. If it is assumed that the mechanical power output from the turbine is essentially constant, the electrical power deviations from the quiescent level will produce a proportional acceleration of the turbo-generator rotors. Thus, if the field voltage is varied in proportion to electrical power deviations (with negative gain), stabilising action is achieved.
8
The investigation and control of the phenomena of poor system damping of oscillation is a complex subject requiring the development of a new approach to power system analysis. The damping level is sensitive to a large range of system parameters, including demand and generation level, generator loading, network configuration and voltage profile.
The primary requirement of excitation control systems is to maintain constant terminal voltage. With the addition of a stabilising signal, this requirement can be relaxed to one of maintaining constant quiescent terminal voltage. The signal processing circuitry for the stabilising signal will depend on the voltage control parameters and the point of stabilising signal injection. The problem of selecting suitable parameters for the overall excitation control system is at present analysed by using a simplified model of a single machine connected through an impedance to a bus which maintains constant voltage and frequency. This degree of simplification is considered necessary because the large number of degrees of freedom involved in system operating conditions makes a complete analysis impractical. The effectiveness of the final parameter selection is examined using the multi-machine eigenvalue program.
Analysis techniques used in these investigations require detailed and flexible machine and excitation system representations. The technique described in this paper utilised a matrix-eigenvalue approach and provides the required level of representation. When used in conjunction with well documented test or system incident information and validated system data, it has been found to be a valuable tool for the determination of limits to system operation and the development of design strategies for improvement of system damping. Improvement of system damping can be achieved through the application of special damping signals to machine excitation systems based on machine output power as a source. Co-ordination is required between the design approach used for such applications and the overall system damping investigations in order to set new limits for the resultant system.
The design technique used is to select the regulator configuration and parameters so that the damping torque (torque in phase with speed variation) is enhanced over the frequency range where system oscillation problems can occur (0.2 Hz to 2 Hz). This has to be done without excessive reduction of synchronising torque (torque in phase with angle variation). The enhancement is required for all generator operating conditions within thermal limits and for a wide range of system impedances - typically 12% to 75% on generator rating.
r ·e r .,
~p~s~~~~
9
,r
..
lD
_____ _
2
A~DIACONO V., et al (1975). Studies and Experimental Results on Electro-mechanical Oscillation Damping in Yugoslav Power System. IEEE PAS SUllllIler Meeting, July 1975.
3
WILSON A., DILLON G., PENFOLD B., BURNS R., GRAINGER A., (1976), Undamped Oscillation in the Interconnected New South Wales, Snowy Mountains and Victorian System. Report to WG03 of CIGRE SP32, May 1976.
4
WHITE, G. (1971). Matrix and Eigenvalue Approach to the Dynamic Stability of MultiMachine Power Systems. Ph.D. Thesis, Melbourne University.
5
SHACKSHAFT G., (1963), General Purpose Turboalternator Model. Proc. lEE Vol. 110, p. 703.
0 -8
0 ·, 04
r O~+____~"=,=z~_ _b ...-""-_----l 0 ·'
O·4+-,-T""T-n''TTIJ--y--,--rr.,.,,+----y---rTT.....,.,.l ·02
· 04 -06 0-1
·2
,4
" ,81-0
2
4
6 8 10
FREQUENCY (~z)
Fig. 4
REFE~CES
LANCE K., BOLDEN R. and SHEARD G. (1975). Performance of Generators with Static Excitation on a Large Network. I.E.Aust. Annual Engi~eering Conference, 1975, pp. 226-235
~ :~+------"...----t-----MIr g
0-01
ACKNOWLEDGEMENTS
The authors wish to acknowledge the contributions made towards this work by fellow engineers in the State Electricity Commission of Victoria, in particular that .iinade by Mr A L Marxsen.
r ·4
~
CONCLUSIONS
Synchronising and Damping Torques as a Function of Frequency for Operation at Rated Power, Unity Power Factor
A typical plot of synchronising and damping torques
53
Investigation and Control of the Damping of Power System Oscillations A .T . WILSON, G.D . WHITE, G.M. BREUER and R.L. BOLDEN State Electricity Commission of Victoria, Melbourne
SUMMARY The development of a design approach to the investigation and control of the damping of power system oscillations is given, including an outline of the analysis technique used and the approach followed in setting limits to system operation. age through control of field voltage, effectively reduce the equivalent damping contribution provided to the system by the machine damper windings for some of these oscillation modes.
INTRODUCTION Limits to the operational flexibility of power systems have classically been defined in terms of power transfer capability between generation and load centres, or between regions of a large interconnected system. In closely coupled systems the limits have been determined by contingent thermal loading of the transmission. However, in weakly coupled systems, the limits have been determined by system phenomena generally categorised in terms of steady state and transient stability of the generators, loads and the transmission network following the loss of the most critical element.
Under certain system conditions these oscillation modes may be excited by the small perturbations that are continually occurring in the system. If the damping associated with a particular oscillation mode is low, the system will oscillate for prolonged periods. Systems characterised by large generation and load centres located electrically at remote ends of a relatively weak transmission system can be particularly prone to this phenomena when transferring power between the centres. Recent examples of this have been experienced in the interconnected systems of Italy and Yugoslavia (ref. 2), and the New South Wales/Snowy/Victoria interconnected systems in A~tralia (ref. 3).
The classical definition of steady state limits have been understood in terms of the 90 degree limit of rotor angle between equivalent generator rotors at either end of the transmission network being studied. In large, multi-machine interconnected systems, however, constraints arising from limited dynamic reactive reserve capability have resulted in a need to re-assess the limits in terms of contingent voltage profile and reactive power losses in the transmission following loss of the most critical element in the network, particularly at periods of high demand in the system.
y
j'
Ensuring ~. adequate understanding of this phenomena and its effect on limits of system operation has required the development, testing, application and interpretation of digital computer based analytical tools. 2
When faults resulting in loss of a major transmission element occur, the fault severity, its duration and the resultant impedance change in the network all contribute to the transient stability performance of the system. Duplicated protection and fast acting circuit breakers have reduced significantly the effect of the fault itself, and the recent development of fast acting excitation systems has further enhanced the overall transient stability performance of the system.
ANALYSIS TECHNIQUES
The normal approach to the study of oscillation damping problems is to use an equivalent linearised system rather than directly solve the basic nonlinear differential equations describing the system. This can then be analysed by classical control system methods, i.e. Nyquist Criterion, Routh Criterion, Root Locus, Domain Separation or Eigenval ue Analysis. The amount of computation required for each method is considerable and it has been common practice to further simplify the analysis by reducing the system to a single machine connected via an impedance to an infinite bus. This simplification is valuable for detailed analysis of an individual machine and regulator, but may not necessarily give an accurate indication of behaviour on a system basis where interactions between machines or groups of machines can occur. It is in this area of system behaviour that a more complete multi-machine analysis is required. For such studies eigenvalue analysis which uses a state space matrix formulation of the equations has computational advantages over the other methods. The use of matrix formulation facilitates the handling of the large number of variables involved and allows the overall system
The need to ensure well-damped and stable operation of the excitation system and synchronous machine When connected to the system has necessitated either a trade off in terms of excitation system response through internal voltage regulator feedback loops or the application of special damping control signals (ref. 1). Further, the power system has a large number of possible natural modes of oscillation associated with the electro-mechanical characteristics of each generator, its associated controls and the electrical parameters of the network connections between them. High gain excitation systems, when connected as a control device regulating generator terminal volt-
49
within the program, they take the form of a series of submatrices down the diagonal with cross coupling terms given by the current equations.
matrix to be built up from sub-matrices describing each section. 2.1
Program Details 2.4
The initial analysis of the oscillation damping problem utilised a small scale multi-machine computer model developed in 1970 as part of a research project (ref. 4). This proved a valuable tool and allowed a good understanding of the phenomena involved, but was somewhat limited due to restriction on size and the relatively simple machine and AVR modelling used. Consequently, a more advanced and sophisticated program has been developed using the same basic approach but with increased size, more flexible modelling and improved input-output features. Details of the main aspects of the program are given below. 2.2
AVR information is expressed in terms of a block diagram. Two block types are recognised as shown in Fig. 1.
A.v'=:0-
Q . - TRANSFER FUNCTlON BLOCK
b .- SUMMER BLOCK
Data Input Fig. 1
Machine Equations
The program processes the AVR data block by block. For each input and output the variables already in the matrix are scanned to locate the variable required. If it is not found the next unused row and column in the matrix is assigned to the variable and the appropriate equation is entered.
MACHINE MODELS
Description
Standard AVR Blocks
By setting one or more of the constants A, B, C, D to zero, the transfer function block can be made to represent a gain, lag block, integrator, etc.
TABLE I
After all the blocks have been processed for one particular AVR, the unwanted (non state variable) variables, including those for the related machine, are eliminated and the matrix compressed.
Flux Variables Included
2.5 1
Vo
C .Ut
The machine representation is based on standard two axis representation (ref. 5). Six machine models of differing complexity and accuracy are available . to allow flexibility of studies (see Table I).
Model
L
II.V1.
A load flow program is used to set up the required operating point and the reduced admittance matrix is produced from a transient stability program. 2.3
AVR Equations
Calculation of Eigenvalues and Eigenvectors
-
Constant voltage behind reactance
2
Id-axis wdg.
1jJ'
3
Id-axis, Iq-axis wdg.
1jJ'd' 1jJ" q
4
2d-axis, Iq-axis wdg.
1jJ'd' 1jJ" d' 1jJ" q
Eigenvalues of the total system matrix are calculated using a standard QR algorithm. Only eigenvalues which are poorly damped are displayed. As an aid to interpretation of result the optional calculation of eigenvectors within a specified frequency band is available.
5
2d-axis, 2q-axis wdg.
1jJ'd' 1jJ" d' 1jJ' q' 1jJ" q
6
Infinite bus representation
3
d
-
VALIDATION OF RESULTS
The following methods of validation are available: 3.1
In the machine equations the effects of stator p1jJ terms and saturation have been neglected because their effect is small for dynamic phenomena. The effect of speed terms in the mechanical and voltage equations have been incorporated.
Comparison with transient (Time-Domain) Program
To properly test the eigenvalue program a full transient stability program including the effect of AVRs must be used; however, the problems associated with separating out the various oscillation modes and calculating their relative damping makes its use of doubtful value.
The effect of governors has also been neglected as their effect on oscillations of the frequencies being considered (0.2 - 2.0 Hz) is negligible. Mechanical damping due to the turbine is allowed for by a speed proportional term in the swing equation. In most studies this speed proportional damping terc is not increased to represent subtransient electrical damping effects because this practice tends to mask the oscillatory phenomena associated with AVR/machine/system interactions.
3.2
Comparison with Single Machine Infinite Bus Model
Various single machine - infinite bus models are available which have been exhaustively tested against each other and against practical tests. Comparison with these programs is valuable for verifying the machine and AVR modelling but does not enable checking of the intermachine behaviour nor an assessment of the effects of assumptions made in setting up the model (e.g. the effects of loads) .
For system studies the machine models used are 2 and 4. Model 2 gives a transient representation, while Model 4 is the normal sub transient representation which includes the effect of damper windings. Neglecting the effect of damper windings gives poor damping on machine local oscillation modes, particularly on synchronous condensers.
3.3
As these equations are placed in the system matrix
Comparison with Actual Power System Performance
If incidents of poor damping oscillations have
50
4.3
occurred, then these can be used as test cases, providing adequate information about the system operating point can be assembled. Although system tests are often difficult to achieve because of the problems of control and instrumentation, they are an essential and integral requirement of program validation.
In this mode, generation areas swing against other generation areas in a loosely coupled mode. An example of this oscillation mode would be Victorian generation swinging as a group against New South Wales generation via the interconnection.
Even with the data from an incident or system test, the eigenva1ue program will normally indicate lower levels of damping than those actually observed. This is due to sources of damping such as machine rotor body effects and the contribution of loads not being explicitly represented in the program. Rather than attempt to represent these effects by the usual speed proportional damping term, results from known system conditions are used to define reference damping levels, which then give the relative increase or decrease in damping for other system conditions. 4
Multi-Machine System Modes
It is in the analysis of multi-machine system modes that the eigenvalue program is most valuable Multi-machine local modes, although of interest, are generally well damped. Single machine local modes can be more conveniently studied on a single machine - infinite bus basis. 5
PRINCIPLES OF INVESTIGATIONS
5.1
Evaluation of Transfer Limits
Oscillation damping is sensitive to power transfers between regions, and it can be the determining phenomena for the limit of transfer. Before an assessment of the transfer limits on the systeu can be undertaken, it is first necessary to identify the system conditions at which poor damping of oscillations is likely to occur. Typically, these conditions are characterised by machines operating heavily loaded and close to their underexcited limits with the system at light load and heavy transfers occurring over long distances or over weak links.
INTERPRETATION OF RESULTS
Interpretation of results from the multi-machine eigenvalue program becomes difficult for high order systems, where up to 20 machines may be represented. As such systems can yie1d,over 100 eigenvalues, it becomes difficult to interpret which particular system components influence particular eigenva1ues. Although the problem is alleviated to some extent by the re~lisation that the system behaviour is primarily governed by lightly damped eigenvalues which are close to the imaginary axis in the eigenvalue plane, there may be several such eigenva1ues, which makes interpretation difficult.
Because of the sensitivity of damping to system operating conditions, it is essential to set up load flows that accurately represent a realistic system. Operating conditions are then studied at various power transfer levels for worst system contingencies, which may be loss of a generator it the importing system leading to an increase in transfer, or loss of a major system interconnecto] A useful analysis aid is to plot system damping (i.e. damping ratio of dominant system eigenvalue) as a ~nction of power transfer, as shown in Figure 3lor the critical contingency.
The problem of interpreting results is made much easier by the calculation of eigenvectors for each of the lightly damped eigenva1ues. This enables the influence of system components on particular eigenvalues to be better understood. In particular, it enables oscillation modes for each machine on the system to be determined. From system studies using this technique, it has been possible to divide machine oscillation modes into three broad categories as shown in figure 2. The influence of a particular machine on a mode is indicated by the magnitude and phase of its eigenvector relative to the eigenvectors of other machines.
OAhIf'ING OF DOMINANT OSCILLATION MODE
+
\'
--- ---
..~ (;l SIHCLE MACltIl4E L.OCAL OSCILLATION IoIOOE
(iil IlULTI-MACHINE LOCAL OSCILLATION MODE
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Fig. 2 4.1
Speed Eigenvectors for Specific Oscillation Modes
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System Damping as a Function of Interchange Between Regions
Single Machine Local Modes Setting up a system study for analysis will usual ] ~equire representation of at least one equivalent machine at each of the major generating stations and representation of equivalent synchronous compensators where these are judged important.
In this mode a single machine oscillates against the system with all other machines contributing very little. This is ana1agous to a single machine oscillating on an infinite bus. This mode tends to exist only for small machines and machines that are remotely situated from other machines. 4.2
FROM REGION
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It is necessary to determine the sensitivity of system oscillatory damping to various system conditions. This sensitivity can then be used to determine constraints that need to be placed on certain system parameters for particular transfer conditions.
Multi-Machine Local Modes
This mode involves one machine group swinging against another machine group where the two machine groups are tightly coupled.
51
5.1.1
5.3
Changes in network structure
The system analysis studies that have been carried out indicate that the accuracy of machine and excitation control system data is especially important for the larger machines on the system because of their dominant contribution to system oscillation modes.
Outages of critical interconnectors have a detrimental effect on system damping to the extent that such outages may significantly reduce transfer capability. Outages of non-critical lines do not have such an effect and in fact may increase system damping at light load if the removal of the additional line charging from the system results in some machines operating in a more stable region. 5.1. 2
6
Possible stabilisation schemes to improve system damping are evaluated. The most cost effective scheme at present is the installation of power stabilisers on appropriate machines on the system. A detailed design of the stabilisation scheme is then undertaken. For the installation of a power stabiliser, this would involve a detailed redesign of the excitation control system on a single machine basis.
Generation distribution
Base load plant remote from load centres is inherently less stable than equivalent generation situated electrically close to the major load centres. This can necessitate the reduction of transfer limits for undesirable generation distributions or the restriction of station output on remote generation for a given transfer limit. 5.1. 4
IMPROVEMENT OF SYSTEM DAMPING
A strategy used for the improvement of system damping is detailed below:
Network loading and distribution
System loading level and load distribution affect the loading of generators and lines. At light system loads, where lines may be well below their surge impedance loading, generators may be required to operate in an under-excited mode to help maintain system voltages at acceptable levels. It is in this operating mode that system damping can become unacceptably low. 5.1. 3
The effectiveness of the stabilisation scheme is then examined using the multi-machine eigenvalue program, for all system operating conditions. The stabilisation scheme is then placed into service and its effectiveness assessed in practice by system tests.
Voltage levels
It has been found that for the region exporting power across an interconnection, the raising of voltage levels at base load power stations is in general stabilising. The effect of changing voltage on the receiving end of the system is not, however, so well defined. 5.1. 5
The multi-machine eigenvalue program is again used to reassess system operating limits with the modified excitation control system in service.
Distribution of inertia 7
System damping is also influenced by system inertia and its distribution. For example, the dominant system oscillation modes will be different for a system where all machines are operating close to full output than for a system of the same total generation with a larger number of machines operating with some machines at reduced output, i.e. low system nett inertia per MW.
7.1
DESIGN OF SYSTEM STABILISING DEVICES Choice of Stabilising Signal
System damping may be enhanced by introducing stabilising signals into the excitation systems of generators. The basic concept is to vary the terminal voltage of generators in phase with speed deviations and so provide damping action.
All the above aspects must be considered when evaluating transfer limits on the system if the advantages of interconnection are to be fully exploited. 5.2
Accuracy of Data
There are three quantities available local to a generator which, in conjunction with suitable signal processing, may be used for stabilising signals. These are generator rotor speed, electrical frequency at the generator terminals and electrical power output. Since, for a given level of system oscillation, the relative deviation in electrical power output is much greater than the relative speed and frequency changes, electrical power is the signal which should present the least problems with quiescent noise from transducers and rotor torsional oscillations. Instantaneous power transducers are readily available and can be installed in existing current and voltage transformer circuits, giving the power signal a distinct advantage over the speed signal, which requires custom built measuring equipment attached to the turbo-generator shaft. Of the three signals available, electrical frequency is the least sensitive to system oscillations and it is difficult to obtain a noise-free frequency signal without introducing significant delays due to filtering circuits. For these reasons, our attention has focussed primarily on stabilising signals derived from electrical power output for introduction into existing excitation systems.
Studying of Incidents
The following procedure has been adopted for analysing oscillation events: Data is collected on system operation conditions just prior to the onset of oscillations. The oscillation event is then simulated using the multi-machine eigenvalue program and the damping of the simulated event is compared to the actual event. The major factors contributing to the poor damping are analysed and interim operating restrictions are defined to ensure that the si tuation does not occur again. If necessary, a follow-up strategy is employed to improve the system damping and extend the allowable range of system operation (see Section 6).
52
7.2
as a function of frequency is given in Figure 4. It can be seen that the stabilised regulator has greatly increased the damping torque in the critical oscillation frequency range.
Stabilisation Using Power Signals
Because of the relatively long time constant of the field of a turbo-generator, the field voltage should be varied in phase with acceleration so that the terminal voltage varies in phase with speed. If it is assumed that the mechanical power output from the turbine is essentially constant, the electrical power deviations from the quiescent level will produce a proportional acceleration of the turbo-generator rotors. Thus, if the field voltage is varied in proportion to electrical power deviations (with negative gain), stabilising action is achieved.
8
The investigation and control of the phenomena of poor system damping of oscillation is a complex subject requiring the development of a new approach to power system analysis. The damping level is sensitive to a large range of system parameters, including demand and generation level, generator loading, network configuration and voltage profile.
The primary requirement of excitation control systems is to maintain constant terminal voltage. With the addition of a stabilising signal, this requirement can be relaxed to one of maintaining constant quiescent terminal voltage. The signal processing circuitry for the stabilising signal will depend on the voltage control parameters and the point of stabilising signal injection. The problem of selecting suitable parameters for the overall excitation control system is at present analysed by using a simplified model of a single machine connected through an impedance to a bus which maintains constant voltage and frequency. This degree of simplification is considered necessary because the large number of degrees of freedom involved in system operating conditions makes a complete analysis impractical. The effectiveness of the final parameter selection is examined using the multi-machine eigenvalue program.
Analysis techniques used in these investigations require detailed and flexible machine and excitation system representations. The technique described in this paper utilised a matrix-eigenvalue approach and provides the required level of representation. When used in conjunction with well documented test or system incident information and validated system data, it has been found to be a valuable tool for the determination of limits to system operation and the development of design strategies for improvement of system damping. Improvement of system damping can be achieved through the application of special damping signals to machine excitation systems based on machine output power as a source. Co-ordination is required between the design approach used for such applications and the overall system damping investigations in order to set new limits for the resultant system.
The design technique used is to select the regulator configuration and parameters so that the damping torque (torque in phase with speed variation) is enhanced over the frequency range where system oscillation problems can occur (0.2 Hz to 2 Hz). This has to be done without excessive reduction of synchronising torque (torque in phase with angle variation). The enhancement is required for all generator operating conditions within thermal limits and for a wide range of system impedances - typically 12% to 75% on generator rating.
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2
A~DIACONO V., et al (1975). Studies and Experimental Results on Electro-mechanical Oscillation Damping in Yugoslav Power System. IEEE PAS SUllllIler Meeting, July 1975.
3
WILSON A., DILLON G., PENFOLD B., BURNS R., GRAINGER A., (1976), Undamped Oscillation in the Interconnected New South Wales, Snowy Mountains and Victorian System. Report to WG03 of CIGRE SP32, May 1976.
4
WHITE, G. (1971). Matrix and Eigenvalue Approach to the Dynamic Stability of MultiMachine Power Systems. Ph.D. Thesis, Melbourne University.
5
SHACKSHAFT G., (1963), General Purpose Turboalternator Model. Proc. lEE Vol. 110, p. 703.
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FREQUENCY (~z)
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REFE~CES
LANCE K., BOLDEN R. and SHEARD G. (1975). Performance of Generators with Static Excitation on a Large Network. I.E.Aust. Annual Engi~eering Conference, 1975, pp. 226-235
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ACKNOWLEDGEMENTS
The authors wish to acknowledge the contributions made towards this work by fellow engineers in the State Electricity Commission of Victoria, in particular that .iinade by Mr A L Marxsen.
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CONCLUSIONS
Synchronising and Damping Torques as a Function of Frequency for Operation at Rated Power, Unity Power Factor
A typical plot of synchronising and damping torques
53