Oscillation analysis method and apparatus therefor

Abstract

The present invention relates to apparatus (10) for determining a contribution of at least one grid subsystem of plural grid subsystems to oscillation in angle or grid oscillation in an electrical grid (12). The apparatus (10) is configured to receive a first quantity which corresponds to oscillation in angle at a first grid subsystem (14). The apparatus (10) is also configured to receive a second quantity which corresponds to oscillation in angle at a second grid subsystem (16). The apparatus (10) comprises a processor (32) which is operative to determine a contribution of at least one of the first and second grid subsystems to oscillation in angle or grid oscillation with the contribution being determined in dependence on a phase relationship between the first and second quantities.

Claims

1. A method of determining a contribution of at least one grid subsystem of plural grid subsystems to oscillation in angle or grid oscillation in an electrical grid, the method comprising: receiving a first quantity which corresponds to oscillation in angle at a first grid subsystem; receiving a second quantity which corresponds to oscillation in angle at a second grid subsystem, the first and second quantities being based on a same measured property; and determining a contribution of at least one of the first and second grid subsystems to oscillation in angle or grid oscillation, the contribution being determined in dependence on a phase relationship between the received first and second quantities, wherein determining the contribution comprises calculating a contribution of at least one of the first and second grid subsystems, wherein the contribution is calculated in dependence on a sine of a difference between the phases of the first and second quantities, and wherein plural such contributions are determined, comparing the plural contributions to thereby determine relative extents of contribution of plural grid subsystems.

2. The method according to claim 1 wherein the same measured property is one of frequency, angle and power.

3. The method according to claim 1 wherein determining the contribution comprises determining relative contributions of the first and second grid subsystems to oscillation in angle, the step of determining relative contributions comprising determining relative damping contributions of the first and second grid subsystems to oscillation in angle.

4. The method according to claim 1 further comprising comparing the contribution of at least one of the first and second grid subsystems with a respective reference contribution.

5. A computer program product comprising program instructions for causing a computer to perform the method according to claim 1.

6. The computer program product according to claim 5 which is one of: embodied on a record medium; embodied in a read only memory; and stored in a computer memory.

7. The method according to claim 1 wherein at least one of the first and second quantities is received by way of processing apparatus from measurement apparatus.

8. The method according to claim 7 wherein the measurement apparatus comprises at least one phasor measurement unit (PMU).

9. A method of determining a contribution of at least one grid subsystem of plural grid subsystems to oscillation in angle or grid oscillation in an electrical grid, the method comprising: receiving a first quantity which corresponds to oscillation in angle at a first grid subsystem; receiving a second quantity which corresponds to oscillation in angle at a second grid subsystem, the first and second quantities being based on a same measured property; and determining a contribution of at least one of the first and second grid subsystems to oscillation in angle or grid oscillation, the contribution being determined in dependence on a phase relationship between the received first and second quantities, wherein determining the contribution comprises determining whether or not there is a change in oscillation in angle, and wherein at least one of temporally spaced apart first quantities corresponding to characteristics of oscillation in angle and temporally spaced apart second quantities corresponding to characteristics of oscillation in angle are received, the change in oscillation in angle being determined in dependence on the received at least one of temporally spaced apart first quantities and temporally spaced apart second quantities.

10. A method of determining a contribution of at least one grid subsystem of plural grid subsystems to oscillation in angle or grid oscillation in an electrical grid, the method comprising: dividing plural grid subsystems into plural groups in dependence on phase relationships of oscillations of the plural grid subsystems; determining a group phase for each of the plural groups; and calculating a contribution of at least one group of the plural groups in dependence on relationships between the group phases of the plural groups, and wherein for at least one of the plural groups: receiving a first quantity which corresponds to oscillation in angle at a first grid subsystem in said at least one of the plural groups; receiving a second quantity which corresponds to oscillation in angle at a second grid subsystem in said at least one of the plural groups, the first and second quantities being based on a same measured property; and determining a contribution of at least one of the first and second grid subsystems to oscillation in angle or grid oscillation, the contribution being determined in dependence on a phase relationship between the received first and second quantities.

11. The method according to claim 10 further comprising determining a contribution of at least one grid subsystem in a group.

12. The method according to claim 10 wherein there is an increase in damping, the method further comprising identifying one of the plural groups as having a greatest increase in phase relative to others of the plural groups with opposing oscillations.

13. The method according to claim 12 wherein the identified group comprises plural grid subsystems, the method further comprising identifying one of the plural grid subsystems within the identified group as having a greatest reduction in phase relative the other grid subsystems in the identified group.

14. The method according to claim 9 wherein the same measured property is one of frequency, angle and power.

15. The method according to claim 9 wherein determining the contribution comprises determining relative contributions of the first and second grid subsystems to oscillation in angle, the step of determining relative contributions comprising determining relative damping contributions of the first and second grid subsystems to oscillation in angle.

16. A computer program product comprising program instructions for causing a computer to perform the method according to claim 9.

17. The method according to claim 10 wherein there are at least three groups, the method further comprising adjusting a difference between group phases between a first group and each of second and further groups in dependence on at least one of: an amplitude for one of the groups; and a sensitivity of the first group to a change in a respective second or further group.

18. The method according to claim 10 wherein the plural grid subsystems are divided into groups in dependence on electrical separateness of the plural grid subsystems.

19. The method according to claim 10 wherein there is a decrease in damping, the method further comprising identifying one of the plural groups as having a greatest reduction in phase relative to others of the plural groups with opposing oscillations.

20. The method according to claim 19 wherein the identified group comprises plural grid subsystems, the method further comprising identifying one of the plural grid subsystems within the identified group as having a greatest increase in phase relative to the other grid subsystems in the identified group.

21. The method according to claim 10 wherein the same measured property is one of frequency, angle and power.

22. A computer program product comprising program instructions for causing a computer to perform the method according to claim 10.

23. Apparatus for determining a contribution of at least one grid subsystem of plural grid subsystems to oscillation in angle or grid oscillation in an electrical grid, the apparatus being configured to: receive a first quantity which corresponds to oscillation in angle at a first grid subsystem; receive a second quantity which corresponds to oscillation in angle at a second grid subsystem, the first and second quantities being based on a same measured property, wherein the apparatus comprises a processor which is operative to determine a contribution of at least one of the first and second grid subsystems to oscillation in angle or grid oscillation, the contribution being determined in dependence on a phase relationship between the first and second quantities, wherein determining the contribution comprises calculating a contribution of at least one of the first and second grid subsystems, wherein the contribution is calculated in dependence on a sine of a difference between the phases of the first and second quantities, and wherein plural such contributions are determined, comparing the plural contributions to thereby determine relative extents of contribution of plural grid subsystems.

24. An electrical grid comprising apparatus according to claim 23.

25. Apparatus for determining a contribution of at least one grid subsystem of plural grid subsystems to oscillation in angle or grid oscillation in an electrical grid, the apparatus being configured to: receive a first quantity which corresponds to oscillation in angle at a first grid subsystem; receive a second quantity which corresponds to oscillation in angle at a second grid subsystem, the first and second quantities being based on a same measured property, wherein the apparatus comprises a processor which is operative to determine a contribution of at least one of the first and second grid subsystems to oscillation in angle or grid oscillation, the contribution being determined in dependence on a phase relationship between the first and second quantities, wherein determining the contribution comprises determining whether or not there is a change in oscillation in angle, and wherein at least one of temporally spaced apart first quantities corresponding to characteristics of oscillation in angle and temporally spaced apart second quantities corresponding to characteristics of oscillation in angle are received, the change in oscillation in angle being determined in dependence on the received at least one of temporally spaced apart first quantities and temporally spaced apart second quantities.

26. An electrical grid comprising apparatus according to claim 25.

27. Apparatus for determining a contribution of at least one grid subsystem of plural grid subsystems to oscillation in angle or grid oscillation in an electrical grid, the apparatus being configured to: divide plural grid subsystems into plural groups in dependence on phase relationships of oscillations of the plural grid subsystems; determine a group phase for each of the plural groups; and calculate a contribution of at least one group of the plural groups in dependence on relationships between the group phases of the plural groups, and wherein for at least one of the plural groups: receive a first quantity which corresponds to oscillation in angle at a first grid subsystem in said at least one of the plural groups; and receive a second quantity which corresponds to oscillation in angle at a second grid subsystem in said at least one of the plural groups, the first and second quantities being based on a same measured property, wherein the apparatus comprises a processor which is operative to determine a contribution of at least one of the first and second grid subsystems to oscillation in angle or grid oscillation, the contribution being determined in dependence on a phase relationship between the first and second quantities.

28. An electrical grid comprising apparatus according to claim 27.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) Further features and advantages of the present invention will become apparent from the following specific description, which is given by way of example only and with reference to the accompanying drawings, in which:

(2) FIG. 1 is a graph of grid frequency over time which shows the effects of grid oscillation;

(3) FIG. 2 is a block diagram representation of apparatus according to the present invention;

(4) FIG. 3A is a flow chart representation of a method according to a first embodiment of the present invention;

(5) FIG. 3B is a flow chart representation of a method according to a second embodiment of the present invention;

(6) FIG. 4 is a plot of damping ratio for an oscillatory mode which has changed from well damped to poorly damped;

(7) FIG. 5 is a plot of the damping contributions of two groups;

(8) FIG. 6 is a plot of the damping contributions of individual grid subsystems;

(9) FIG. 7 is a compass plot for an inter-area mode;

(10) FIG. 8 is a compass plot for a well damped inter-area mode;

(11) FIG. 9 is a compass plot for a lightly damped inter-area mode;

(12) FIG. 10A is a vector plot of oscillations for an un-damped mode;

(13) FIG. 10B is a vector plot of oscillations for a damped mode;

(14) FIG. 10C is a vector plot of oscillations for a negatively damped mode;

(15) FIG. 11A is a vector plot for two generators with identical damping;

(16) FIG. 11B is a vector plot for two generators with one generator only providing damping; and

(17) FIG. 11C is a vector plot for two generators with both generators oscillating in the same direction.

DESCRIPTION OF EMBODIMENTS

(18) A graph of grid frequency over time which shows the effects of grid oscillation is shown in FIG. 1. FIG. 1 is described above.

(19) A block diagram representation of apparatus 10 according to the present invention is shown in FIG. 2. The apparatus 10 comprises an electrical grid 12 which is operative at power transmission system voltage levels, such as at 132 kV, 220 kV or 400 kV or perhaps less commonly at 750 kV. First, second, third and fourth grid subsystems 14, 16, 18, 20 are electrically connected to the electrical grid 12. Each of the grid subsystems 14, 16, 18, 20 contributes electrical power to the grid and thus comprises a generator and whatever further local apparatus may be required, such as a step-up transformer. Alternatively one or more of the grid subsystems 14, 16, 18, 20 is a section of the grid comprising plural generators and loads that are electrically interconnected. First, second, third and fourth Phasor Measurement Units (PMUs) 22, 24, 26, 28 are provided at a respective one of the first to fourth grid subsystems 14, 16, 18, 20 whereby a PMU is operative to make measurements at or near the point of connection of its respective grid subsystem to the electrical grid 12. Each of the PMUs is an Alstom P847 from Alstom Grid of St. Leonards Avenue, ST17 4LX, Stafford, United Kingdom. The PMUs are synchronised with one another either of themselves or by way of an external time reference, such as from a GPS time source. The apparatus further comprises computing apparatus 30, which in turn comprises a processor 32, data storage 34 and an output device 36. The computing apparatus 30 and its components will be of a form and function familiar to the skilled reader. The output device 36 of the computing apparatus 30 is operative under control of the processor 32 to display data to a user of the computing apparatus 30. The computing apparatus 30 is operative to receive measurements made by the first to fourth PMUs 22, 24, 26, 28. Measurements are received by the computing apparatus 30 by way of a communications channel between the computing apparatus 30 and each PMU with the communications channel being of a copper, optical fibre or wireless form.

(20) The operation of the apparatus 10 of FIG. 2 will now be described with reference to flow charts shown in FIGS. 3A to 3B. FIG. 3A is a flow chart representation 50 of how determinations are made with regards to relative damping contributions where there are two groups of generators oscillating in accordance with an inter-area mode. As is described above the inter-area mode is characterised by opposing oscillations, i.e. oscillations which are substantially 180 out of phase. As a first step 52, first quantities are received in the computing apparatus 30 from a first group which consists of first and second generators comprised in the first and second grid subsystems 14, 16. As a second step 54, second quantities are received in the computing apparatus 30 from a second group which consists of third and fourth generators comprised in the third and fourth grid subsystems 18, 20. The first and second quantities are based on a same or different measurable property such as angle or frequency. Furthermore the quantities, whether they are based on the same or different measurable properties, correspond to oscillations in angle as reflected in the measurable property. Calculation of the quantities on the basis of actual measurements made by the PMUs is normally in a processor 32 in the form of a Wide Area Monitoring System (WAMS) on the basis of unprocessed measurements received from the PMUs. The processor is operative on the PMU measurements to provide oscillation frequency, oscillation amplitude, damping and phase. The determination of these quantities will be familiar to the notionally skilled reader and is described in IEEE Task Force Report TP462 Identification of Electromechanical Modes in Power Systems, June 2012. Alternatively but less frequently the quantities are calculated in the PMUs themselves. The next step 56 involves determining a phase relationship amongst the first and second quantities which is carried out in the processor 32. FIG. 7 is a compass plot for the four generators for a specific frequency of oscillation. As can be seen from FIG. 7 the first group of first and second generators lags the second group of third and fourth generators by an angle of less than the ideal case of 180. The processor 32 is operative to determine this phase relationship. The next step 58 involves determining the relative damping contributions on the basis of the determined phase relationship. The lagging of the first group by the second group by an angle of less than 180 is indicative of the first group contributing more damping than the second group. Accordingly the processor 32 is operative on the basis of the determined phase relationship to determine that the first group contributes more damping and the second group contributes less damping. Considering the first group more closely and as shown in FIG. 7 the first generator lags the second generator which indicates that the first generator contributes the more damping within the first group. The processor 32 is accordingly operative on the determined phase relationships to determine that of the first and second generators the first generator contributes the more damping. Where the electrical grid 12 is exhibiting undesirable oscillatory behaviour, an operator can then address the problem in view of the determination made with regards to relative damping contributions. For example the operator can then take action by adjusting dispatched operating points or by isolating faulty equipment on the basis of the determined relative damping contributions. As will be appreciated, under certain circumstances relative damping contributions of generators can be determined in a straightforward manner on the basis of their phase relationship, such as when generators in a group are oscillating in the same direction. Otherwise a more sophisticated approach is required to take account of more complex oscillatory behaviour which is characteristic of the like of inter-area and local modes. Such a more sophisticated approach is described below with reference to FIG. 3B. Furthermore and according to certain embodiments a more precise determination of damping contributions can be made where there is further characterisation of the grid subsystems and the electrical grid.

(21) FIG. 3B is a flow chart representation 70 of how determinations are made with regards to damping contributions amongst plural generators. As a first step 72 a change in damping is detected. The change in damping is detected on the basis of measurements made in the electrical grid 12 over time. The measurements should provide an indication of overall system behaviour and are, if appropriate, provided by one or more of the first PMUs 22, 24, 26, 28. Alternatively one or more measurements elsewhere within the electric grid 12 are used to detect a change in damping. FIG. 4 shows a significant change in damping ratio within an electrical grid from a well damped condition to a lightly damped condition. If there has been a change in oscillatory behaviour that warrants intervention the oscillatory mode is then determined. Therefore according to a next step 74, system behaviour is analysed to determine whether or not there is common mode oscillation. According to a first approach measured oscillation frequency is compared to a first threshold frequency of 0.1 Hz and if the measured oscillation frequency is less than the first threshold frequency common mode oscillatory behaviour is determined. According to a second approach amplitudes of oscillation in different parts of the electrical grid are compared and if the amplitudes are substantially or generally the same common mode oscillatory behaviour is determined. If common mode oscillatory behaviour is determined the method proceeds to step 84, which is described below. Otherwise the next step 76 is analysis of system behaviour to determine if oscillations in angle are opposing, e.g. as is characteristic of a local mode or an inter-area mode. According to a first approach measured oscillation frequency is compared to a second threshold frequency of 0.2 Hz and if the measured oscillation frequency is more than the second threshold frequency, opposing oscillatory behaviour is determined. According to a second approach amplitudes of oscillation in different parts of the electrical grid are compared and if the amplitudes are significantly different opposing oscillatory behaviour is determined.

(22) If opposing oscillatory behaviour is determined the next step 78 is dividing the grid subsystems or generators into groups. According to a first approach the grid subsystems are divided into groups in dependence on their electrical separateness. More specifically this involves analysing measurements, such as current measurements, made in the electrical grid, for example by the first to fourth PMUs 22, 24, 26, 28 and/or other measurement apparatus operative to make measurements elsewhere in the electrical grid. According to a second approach the grid subsystems are divided into groups in dependence on the relative directions of oscillations of the grid subsystems. More specifically the second approach involves determining a direction of oscillation of each grid subsystem and dividing the plural grid subsystems into groups in dependence on the determined directions. The directions of oscillation are determined by analysis of oscillation measurements, such as measurements made by the first to fourth PMUs 22, 24, 26, 28 and/or other measurements made by measurement apparatus elsewhere in the electrical grid. In certain embodiments the first and second approaches are used together. According to such embodiments the second approach is used initially and the first approach is then used, if necessary, to address the like of conflicts or ambiguities arising from use of the second approach. A further alternative approach involves grouping grid subsystems in dependence on knowledge, e.g. in the form of machine readable data, of the configuration of the electrical grid.

(23) When the grid subsystems have been grouped, the next step 80 is determination of an average phase angle for each group. The average phase angle is determined on the basis of the received quantities (as per boxes 52 and 54 in FIG. 3A) for the grid subsystems or generators comprised in each group. Where ratings for generators within a group differ significantly, the average phase is weighted in dependence on the ratings of the individual generators. Thereafter the damping contribution for each group is calculated 82. The damping contribution for each group is given by:

(24) $D_{\mathrm{Gi}}=\underset{j=1}{\overset{n}{.Math.}}{w}_{\mathrm{ij}}*a_{\mathrm{Gj}}*\sin \left({}_{\mathrm{Gj}}-{}_{\mathrm{Gi}}\right)$
where D.sub.Gi is the damping contribution of group i, w.sub.ij is the sensitivity of group i to group j, a.sub.Gj is the average amplitude for group j, .sub.Gj is the average oscillation phase angle for group j and .sub.Gi is the average oscillation phase angle for group i. The derivation of and basis for the above equation is provided in the Appendix below. Considering the above equation further, the average phase angle for appropriate groups is provided by the previous step. The average amplitude for a group is determined on the basis of amplitude measurements provided by the appropriate PMUs. The sensitivity factor w.sub.ij is determined in dependence on electrical distance between the two groups in question and the ratings of the generators. According to another approach the sensitivity factor is determined by way of a model of the power system. For example and in a power system comprising first to fourth generators, sensitivity factors for the first generator w.sub.i1 are determined by increasing the output of the first generator by a small amount, such as 1 MW, and calculating the changes in the other generators by way of the model, with each of the three sensitivity factors w.sub.i1 being given by

(25) $-\frac{P_i}{{}_1}.$
The same process is then performed in respect of each of the second to fourth generators to thereby provide a set of three sensitivity factors for each of the second to fourth generators. Where required by circumstances, the same approach is applied to groups of generators by modelling each group as one large generator. When an inter-group sensitivity is changed on account of a change in the configuration of the electrical grid the appropriate factor is changed. Upon the conclusion of the present step 82 the damping contributions of the groups are analysed to identify groups which are contributing in a positive fashion to grid oscillation, contributing in a negative fashion to grid oscillation and merely responding to grid oscillation. FIG. 5 is a plot of the damping contributions of two groups which reflect the reduction in damping evident from FIG. 4. As can be seen from FIG. 5 there is a reduction in damping contribution from group 1 and an increase in damping contribution from group 2 whereby it is determined that group 1 is responsible for the reduction in damping seen in FIG. 4.

(26) The following step 84 comprises calculating the damping contributions within at least one of the groups formed at step 78. Depending on circumstances and requirements, damping contributions within a dominant group only are calculated. In other circumstances damping contributions within several groups are calculated. As mentioned above, where a determination of common mode oscillation is made in step 74 of the method the next step is the present step. Where a determination of common mode oscillation is made all the grid subsystems or generators are considered as belonging to one group. In the present step 84 the damping contribution for grid subsystem or generator within the group is calculated on the basis of:
D.sub.i=sin(.sub.G.sub.i)
where D.sub.i is the damping contribution at grid subsystem i, .sub.i is the phase angle of oscillations at grid subsystem i and .sub.G is the average oscillation phase angle for the grid subsystems within the group. Damping contributions within a group and/or at group level are displayed to an operator by way of the output device 36 of the computing apparatus 30 in a numerical or graphical form so as to provide for ease of interpretation of contributory behaviour. FIG. 6 is a plot of the damping contributions of individual grid subsystems within group 1 of FIG. 5 which reflect the reduction in damping evident from FIG. 4. As can be seen from FIG. 6 measurement 1, which is made in respect of a first generator, shows a large reduction in damping contribution whereas the other measurements show a small increase in damping contribution whereby it is determined that the first generator is responsible for the reduction in damping seen in FIG. 4.

(27) Considering the analysis of damping contributions at the group level, where there is an increase in damping, the group having the greatest increase in phase relative the others groups (i.e. involving anti-clockwise rotation relative the other groups) is identified. The identified group is responsible or is the most responsible for the increase in damping. Group contributions are compared with reference contributions which are either based on earlier measurements or reference conditions, such as are provided by simulation or calculation to reflect ideal, well damped conditions. Normally the latter approach is preferred because it tends to yield better results on account of a greater extent of difference and perhaps also additional useful data depending on the nature of the reference conditions. In certain embodiments this approach is expanded to identify a second group having a second greatest increase in relative phase and so on whereby contributions from several groups and their relative extent of contribution are determined. Turning now to the grid subsystem or generator level, the grid subsystem within a particular group which has the greatest decrease in phase relative the other grid subsystems is identified. Contributions at grid subsystem level are compared with reference contributions. The identified grid subsystem within the identified group is responsible or is the most responsible for the increase in damping. Similarly in certain embodiments this approach is expanded to identify a second grid subsystem having a second greatest decrease in relative phase and so on whereby contributions from several grid subsystems and their relative extent of contribution are determined. The effect at the group level is illustrated by way of FIGS. 8 and 9 which show compass plots for an inter-area mode, which changes from the well damped condition reflected in FIG. 8 to a lightly damped condition reflected in FIG. 9. As can be seen, the group with oscillations near 0 is responsible for the reduction in damping as the group which shows a reduction in phase angle.

(28) Considering further the analysis of damping contributions at the group level but now where there is decrease in damping, the group having the greatest reduction in phase relative the others groups (i.e. involving clockwise rotation relative the other groups) is identified. The identified group is responsible or is the most responsible for the decrease in damping. In certain embodiments this approach is expanded to identify a second group having a second greatest reduction in relative phase and so on whereby contributions from several groups and their relative extent of contribution are determined. Turning now to the grid subsystem or generator level, the grid subsystem within a particular group which has the greatest increase in phase relative the other grid subsystems is identified. The identified grid subsystem within the identified group is responsible or is the most responsible for the decrease in damping. Similarly in certain embodiments this approach is expanded to identify a second grid subsystem having a second greatest increase in relative phase and so on whereby contributions from several grid subsystems and their relative extent of contribution are determined.

(29) Returning to FIG. 3B control of one or more grid subsystems is then effected 86, such as by way of adjusting dispatched operating points or by isolating faulty equipment, to address problematic oscillatory behaviour in the electrical grid. Alternatively the present invention is applied to provide feedback on the effect of a control strategy. For example the damping effect of a generator may be increased and reduced by tuning of its controller and the wider effect of the tuning on system stability determined by way of the present invention.

(30) Appendix

(31) A second order system is the simplest dynamic system that can have oscillations.

(32) The behaviour of a second order system can be described by the following differential equation:
{umlaut over (x)}+2.sub.n{dot over (x)}+.sub.n.sup.2x=0(a)
where x is displacement, .sub.n is speed and is the damping ratio.

(33) A generator-infinite bus system can be approximated by a second order model. Here x is the generator angle . Assuming constant mechanical power, the second order model can be expressed as:

(34) $\begin{array}{cc}\stackrel{\mathrm{.Math.}}{}=\frac{1}{2H}\left(-P\right)& \left(b\right)\end{array}$
where H is the inertial constant and P is power.

(35) If the voltage is constant and the generator has no damping P is proportional to the change in the generator angle as defined by:

(36) $\begin{array}{cc}P=V_1{V}_2\frac{\cos {}_0}{X}& \left(c\right)\end{array}$
where V.sub.1 and V.sub.2 are voltage signals on the two buses and X is impedance.

(37) In the present case the eigenvalue has a zero real part.

(38) To find the phase relationship between angle and speed oscillations we assume the angle oscillation is given by the following equation:
x=e.sup.t(d)
where is the eigenvalue. The derivative of this equation is shifted from x by an angle equal to the angle of the complex number . The derivative is given by the following equation:
{dot over (x)}=e.sup.t(e)

(39) Where the eigenvalue has a zero real part, =j.sub.n and {dot over (x)} leads x by 90.

(40) For a damping ratio of 20% the derivative leads by 101. For most observable oscillation modes the angle difference deviates little from 90.

(41) For a single generator the speed, power and angle oscillations can be represented by vectors as shown in FIG. 10A. In this case power P and the generator angle are in phase, as expected from equation (c) and both lag the speed by 90. For a damped mode where damping is provided by electrical power, the electrical power has a component in phase with the speed as shown in FIG. 10B. The damping power component Pd may, for example, result from the generator's voltage controller. The phase relationship for a negatively damped mode where Pd is in the opposite direction is shown in FIG. 10C.

(42) In a system with two generators the power of each generator is a function of the angle of both generators and more specifically of the difference between the two angles. As is shown in FIG. 11A the oscillations are 180 out of phase for two generators with identical damping. If one of the generators has a larger damping component the generator with the greater damping contributes positive damping to the other generator. The generator with less damping has a negative effect on the greater damped generator. As a consequence the damping of the mode is the same for the whole system on account of the interaction between the two generators. This causes the phase shift between the oscillations to differ from 180 as is shown in FIG. 11B.

(43) A system having two generators which are oscillating in the same direction is shown in FIG. 11C.

(44) In both FIGS. 11B and C generator 1 contributes more damping than generator 2. As a result of the phase angle differences the angle oscillations in generator 2 have a component which is perpendicular to the oscillations in generator 1. This perpendicular component is proportional to the sine of the angle difference and causes an increase in the damping contribution of generator 1 compared to the case of FIG. 11A where there is equal damping. This can be expanded to cover a larger number of generators by adding the effect of all generators on generator 1 to thereby calculate the increase in damping provided by a generator compared to the reference case where all generators provide the same amount of damping. The damping contribution for each generator is defined as:

(45) $\begin{array}{cc}{D}_i=\underset{j=1}{\overset{n}{.Math.}}{c}_{\mathrm{ij}}*a_j*\sin \left({}_j-{}_i\right)& \left(f\right)\end{array}$
where D.sub.i is the damping contribution of generator i, a.sub.l is the amplitude of the angle oscillations at generator j, .sub.j is the phase angle of the oscillations at generator j, .sub.i is the phase angle of the oscillations at generator i and c.sub.ij is the reduction in the power output of generator i for a small increase in the angle at generator j, where

(46) $c_{\mathrm{ij}}=-\frac{P_i}{{}_j}.$