Compensation of power oscillations in a power grid

Abstract

Method for compensating for undesired power oscillations propagating throughout a power grid, the method comprising: obtaining phasor angle measurements of an electrical metric characterising said undesired power oscillations at consecutive time instances; determining, based on the measurements, a system matrix; performing an eigenvalue decomposition of the system matrix, thereby obtaining complex eigenvectors and complex eigenvalues; selecting a dominant eigenvalue associated to an undesired power oscillation and a corresponding dominant eigenvector; identifying two furthest entries in the dominant eigenvector having a largest difference of arguments among all entries in the dominant eigenvector; projecting entries of the dominant eigenvector in a complex plane along directions defined by the furthest entries, resulting in a projected eigenvector having complex projected entries, characterising compensating phase angles of the undesired power oscillation; determining, in the power grid, respective time delays from the compensating phase angles for compensation of the undesired power oscillation.

Description

COMPENSATION OF POWER OSCILLATIONS IN A POWER GRIDTechnical Field

[0001] Various example embodiments relate to a method, a data processing system, a computer program and a computer-readable medium for compensating for undesired power oscillations throughout a power grid.Background

[0002] A grid, power grid, or power transmission system comprises a collection of generators, able to generate power, and loads, able to consume power, interconnected through a mesh of transmission lines spread over a geographical region. Operation of a grid is managed by controllers that are installed throughout the power system, e.g. in the generators and loads.

[0003] Power generators operate coordinately to allow control over the synchronous operation of the power grid and to avoid malfunctions, such as a generator disconnection or a black-out. Power oscillations may cause sub-optimal performance of the power grid and may even result in malfunctions, which may cause enormous damage to the power grid infrastructure. Therefore, oscillations of power-related electrical quantities, e.g. power, voltage or current, between power generators are to be controlled. The grid synchronicity can be lost when the voltage phasor angle of the power generators in the grid have a high phase difference.

[0004] Synchronicity is affected by delays in power propagation introduced by components of the power system, such as transmission lines and controllers. In addition, generators may deliberately introduce delays to deliver a specified power into the power grid. Also transient events, such as connecting a load or disconnecting a transmission line, may introduce power oscillations. Furthermore, phase differences increase as transferred power within the grid increases. The combined effect of all non-ideal behaviour can be observed as undesired power oscillations within the grid. Such power oscillations may appear as low frequency, sub-synchronous power oscillations.Summary

[0005] According to some aspects, there is provided the subject matter of the independent claims. Some further aspects are defined in the dependent claims.

[0006] According to a first example aspect, a method is provided for compensating for undesired power oscillations propagating throughout a power grid. The method comprising a step of obtaining, from N power measurement systems, PMSs, in the power grid, phasor angle measurements of an electrical metric characterising said undesired power oscillations at M consecutive time instances. The method further comprises a step of determining, based on the measurements, a linear approximation of dynamics among the phasor angles of the N PMSs, thereby obtaining a system matrix. The method further comprises a step of performing an eigenvalue decomposition of the system matrix, thereby obtaining N complex eigenvectors and N complex eigenvalues. The method further comprises a step of selecting a dominant eigenvalue associated to an undesired power oscillation at an angular frequency and a corresponding dominant eigenvector. The method further comprises a step of identifying two furthest entries in the dominant eigenvector having a largest difference between arguments among all entries in the dominant eigenvector. The method further comprises a step of projecting entries of the dominant eigenvector in a complex plane along directions defined by the furthest entries, resulting in a projected eigenvector having N complex projected entries, characterising compensating phase angles of the undesired power oscillation in the N PMSs. The method further comprises a step of determining, for one or more power compensating units, PCUs, in the power grid, one or more respective time delays from the N compensating phase angles for compensation of the undesired power oscillation at the angular frequency.

[0007] The power grid, also referred to as grid or power system, comprises a plurality of N power measurement systems, PMSs, N denoting the number of PMSs and being a natural number larger than zero. A power measurement system, PMS, is a device configured to measure a phase angle, also referred to as phasor angle orphase, of an electrical metric or quantity, e.g., voltage or current. These can be represented as a phasor according to their sinusoidal character. Such a PMS may, for example, be a phasor measurement unit, PMU, also referred to as phase measurement unit. The PMSs are geographically spread over the grid, allowing collection of measurements at different locations. The measurements are conducted in the PMSs during a common time interval. This allows comparing different PMS results at a specific time instance within the common time interval. Optionally, interpolation techniques may be used within the common time interval. The common time interval may be characterised by a plurality of M consecutive time instances, M denoting the number of time instances and M being a natural number larger than zero. The measurements can be represented by an N-by-M data matrix, wherein each row holds a chronological sequence of measurements obtained in one of the PMSs at the M time instances. Each column of the data matrix holds the measurements obtained at one of the time instances in each of the N PMSs. Power oscillations can be observed as electromechanical waves that experience phase shifts as they propagate through different paths of the grid. Therefore, the measurements capture the undesired power oscillations due to the spatial information stemming from the locations of the PMSs.

[0008] After collecting the measurements, data-driven system identification is performed, resulting in an N-by-N system matrix. The system matrix characterises the undesired power oscillations occurring within the common time interval.

[0009] An eigenvalue of the system matrix represents a contribution of a specific power oscillation. Each entry, also referred to as element, of a corresponding eigenvector represents the contribution of each of the PMSs to a specific power oscillation characterized by a given angular frequency. As such, an eigenvector embodies the collective influence of all the PMS on a distinct power oscillation.. In general, eigenvalues and eigenvectors can comprise complex numbers. After selecting an undesired power oscillation to be mitigated, the corresponding dominant eigenvalue and dominant eigenvector can be identified.

[0010] Generators or groups of generators may oscillate against each other under certain conditions, for example, when they are interconnected by long transmissionlines. This can be observed when a phase offset of approximately 180 degrees occurs between the oscillating generators or groups of generators. As an example, this may occur between generators located at outer edges of the grid. Phase offset may also be referred to as a phase shift or phase lag. The phase offset closely approximates 180 degrees when an oscillation occurs between distant areas that are connected through long transmission lines. The entries of the dominant eigenvector are complex numbers, each representing a contribution at one of the PMSs to the selected undesired power oscillation. Therefore, by identifying the entries having a largest difference, in absolute value, between arguments, wherein the difference of arguments is defined in the interval [-180°, +180°], the PMSs are identified that are closest to the outer borders where generators are oscillating against each other. In the complex plane, each of the entries of the dominant eigenvector is individually projected along the directions, defined within the complex plane, of these furthest entries, according to the method. This causes a change of bases to a new coordinate system wherein the axes are opposite to each other. The entries are rotated such that the projected entries become located within the first and second quadrants of the complex plane. By doing so, the projected entries are stretched out to cover a span of 180 degrees in the complex plane, thereby revealing the phase angles that would compensate the undesired power oscillation in each PMS. Thus, the method may provide undesired power oscillation compensation by exploiting the counter-phase behaviour of oscillations over geographically spread locations.

[0011] The compensating phase angles are related to physically compensating time delays by the angular frequency of the undesired power oscillation. The power grid comprises one or more power compensating units, PCUs. A PCU is a device configured to inject power into the power grid or to absorb power from the power grid. The PCUs may, for example, be power conversion stations, also referred to as power converters. A time delay is calculated for each of the PCUs, such that, when each of the PCUs applies its compensating time delay simultaneously, the undesired power oscillation is damped throughout the grid.

[0012] The method may provide cancelling out, or counteracting, of undesired power oscillations by applying the compensating time delays. As such, power propagationdamping may be achieved. By taking into account the actual phase shifts occurring in the power grid, critical phase shifts can be easily monitored and avoided. The method may further be applied repeatedly to keep adjusting to the undesired power oscillations that may change over time. As such, the method allows detecting and compensating for real-time power oscillations, and allows handling transient or asynchronous behaviour. By performing damping based on data-driven techniques, the method is adaptable to unpredictable circumstances, as opposed to modelbased approaches. Thereby, the method may reduce a probability of defects. In addition, the data-driven approach introduces limited calculation costs, enabling quick evaluation and oscillation compensation. Further, the efficient damping of undesired power oscillations may reduce energy losses and may provide a higher power transfer, which may lead to reduced operational costs. A further advantage is that the undesired power oscillations may be corrected in PCUs of existing infrastructure. The time delays may be calculated for any location in the power grid, irrespective of the locations of the PMSs where the data is obtained from. The time delay compensation enables damping by active as well as reactive power modulation, thereby allowing tackling of both active and reactive power individually.

[0013] The method allows power plants to conform with regulations and restrictions to be met to connect to a grid by allowing them to contribute to damping. The method may be applied in complex grids, regardless of renewable energy complicating grid management. Large-scale wind and solar farms can respond more quickly than traditional electromechanical generators, such as a turbine of a nuclear plant, and may cause more significant power oscillations. In addition, with renewable energy, customers may inject energy back into the power grid, e.g. when solar panels generate excess energy that is not needed by a customer and that cannot be stored. The method may even take advantage of this bidirectional energy flow, by treating customers as PCUs that can compensate undesired power oscillations by applying corresponding time delays.

[0014] According to further example embodiments, the determining of the system matrix comprises performing a dynamic mode decomposition, DMD, algorithm.

[0015] The DMD algorithm extracts two N-by-M-1 matrices from the N-by-M data matrix. A first data matrix is obtained by leaving out the last column and a second data matrix is obtained by leaving out the first column. As such, the first and second data matrices comprise temporally subsequent measurements. The DMD algorithm finds the system matrix as the best fitted N-by-N matrix that linearly relates the first and second data matrices. Thereby, the behaviour of the system from one time instance to the next is captured.

[0016] Deploying the DMD algorithm allows efficiently computing the system matrix in a limited number of computations.

[0017] According to further example embodiments, the determining of the system matrix comprises performing: a proper orthogonal decomposition, an eigenvalue realisation algorithm, also referred to as ERA, the Koopman operator theory for dynamical systems, wherein the system matrix corresponds to the Koopman operator, or numerical algorithms for subspace state space system identification, also referred to as N4SID.

[0018] According to further example embodiments, the selecting of the dominant eigenvalue comprises selecting the eigenvalue corresponding to a most critical undesired power oscillation.

[0019] Although the method may be performed subsequently for each of the undesired power oscillations, it may be most efficient to focus on the undesired power oscillation that corresponds to the largest contribution of all the eigenvalues.

[0020] According to further example embodiments, the selecting of the dominant eigenvalue comprises selecting the eigenvalue having the smallest absolute value of its real part.

[0021] Each eigenvalue is associated to an undesired power oscillation that is characterised by a certain angular frequency and a certain damping factor. The imaginary part of an eigenvalue of the system matrix may correspond to the angular frequency of the oscillation corresponding to that eigenvalue. The normalised real part of an eigenvalue of the system matrix may correspond to the damping factorthat applies for the oscillation corresponding to that eigenvalue. With normalised real part is meant the real part divided by the magnitude of the complex eigenvalue.

[0022] By selecting the eigenvalue having the smallest absolute value of its real par, or, , the smallest damping factor, the power oscillation is selected that is the least damped by the power grid. Advantageously, by selecting the power oscillation that is least damped, the power oscillation that may have the longest duration before diminishing is compensated.

[0023] According to further example embodiments, the selecting of the dominant eigenvalue comprises selecting the eigenvalue having a largest magnitude.

[0024] According to further example embodiments, the selecting of the dominant eigenvalue comprises selecting the eigenvalue having an imaginary part closest to a predetermined angular frequency, thereby targeting a specific undesired power oscillation corresponding to the predetermined angular frequency.

[0025] A specific angular frequency may be known to degrade the operation of the power grid. For example, due to interference with signals purposely having that angular frequency. This may cause problems even though it may not be the angular frequency providing the largest contribution to the undesired power oscillations. In the selecting step of the method, power oscillation components at any predefined frequency may be compensated.

[0026] According to further example embodiments, the determining of the time delays comprises scaling the projected entries with an inverse of the angular frequency, wherein the inverse is the multiplicative inverse.

[0027] A phase angle, expressed in radians or degrees, is related to a time delay, expressed in seconds, by a factor equal to the angular frequency, expressed in radians or degrees per second.

[0028] According to further example embodiments, the determining of a time delay in a PCU and / or PMS comprises interpolating compensating phases in nearby PMSs based on a distance to the nearby PMSs.

[0029] PMSs may coincide with PCUs, but this is not required. The time delays are determined from the perspective of the PMSs, since the measurements at the locations of the PMSs are used as an input for the method. Phases and time delays relate to distance, and may even be approximated to relate linearly to distance. Therefore, by applying interpolation the time delays can be determined for PCUs at different locations from the PMSs’ locations. Extrapolation techniques may also be applied, e.g. in conjunction with interpolation. Advantageously, the method does not impose any requirements on the spatial locations of the PMSs and of the PCUs, but has the flexibility to be effective regardless the exact locations of the PMSs and of the PCUs.

[0030] According to further example embodiments, the method further comprises, by at least one of the PCUs, performing the steps of: determining an active power steering signal based on a derivative of phasor angle measurements in the at least one PCU; applying a band pass filter, having a centre frequency equal to the angular frequency of the undesired power oscillation, to the active power steering signal; applying the time delays for the at least one PCU to the filtered active power steering signal; and performing active power injection based on the delayed active power steering signal.

[0031] According to further example embodiments, the method further comprises, by at least one of the PCUs, performing the steps of: determining a reactive power steering signal based on a derivative of power transfer in the at least one PCU; applying a band pass filter, having a centre frequency equal to the angular frequency of the undesired power oscillation, to the reactive power steering signal; applying the time delays for the at least one PCU, offset by 90 degrees, to the filtered reactive power steering signal; and performing reactive power injection based on the delayed reactive power steering signal.

[0032] The method may further comprise determining active and reactive power reference signals to be used to regulate power output of the PCUs. Physical distances between oscillating sources translate to electrical distances causing unwanted active and reactive power modulation. Active power compensation may be most efficient close to an oscillation source, while reactive power compensationmay be most efficient near the centre of long power transmission lines, or, in other words, furthest away from oscillating sources. The derivative of the voltage angle is the highest closest to the oscillation source. The derivative of power transfer, i.e. active power output vs. active power input, is higher closer to the middle, i.e. further away from the oscillation source. Therefore, the derivative of phasor angle measurements in the PCU may be advantageously chosen as the active power steering signal; and the derivative of power transfer may be advantageously chosen as the reactive power steering signal. The filtering is applied to the power steering signals to obtain the oscillatory component corresponding to the selected dominant oscillation. Next, the corresponding time delay is applied to compensate for the undesired power oscillation. The filtered and delayed power steering signals may then be used as a power reference for power injection into the grid. In the case of reactive power, the delay needs to be offset by 90 degrees in order to effectively inject reactive power into the grid. Without applying the offset of 90 degrees, power will be injected as active instead of reactive power.

[0033] According to further example embodiments, the method further comprises the steps of: obtaining, from the N power measurement systems, PMSs, in the power grid, phasor amplitude measurements of the electrical metric at M consecutive time instances; determining, based on the phasor amplitude measurements, a damping distribution of the power grid; and determining, for one or more of the PCUs, one or more respective compensating gains based on the damping distribution for amplitude compensation of the undesired power oscillation at the angular frequency.

[0034] In addition to the phasor angle measurements, phasor magnitude or amplitude measurements may be taken into account to provide for an even more effective damping.

[0035] According to a second example aspect, there is provided a data processing system comprising means for carrying out the method according to the first example aspect. Such a data processing system may provide one or more of the above- mentioned advantages.

[0036] According to a third example aspect, there is provided a computer program comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method according to the first example aspect. Such a computer program may provide one or more of the above-mentioned advantages.

[0037] According to a fourth example aspect, there is provided a computer-readable medium comprising instructions which, when executed by a computer, cause the computer to carry out the method according to the first example aspect. Such a computer-readable medium may provide one or more of the above-mentioned advantages.Brief Description of the Drawings

[0038] Fig. 1 shows a schematic view of a power transmission grid;

[0039] Fig. 2 shows a schematic view of a method for compensating undesired power oscillations in a power grid according to example embodiments;

[0040] Fig. 3 shows a schematic view of steps of a method for compensating undesired power oscillations in a power grid according to example embodiments;

[0041] Fig. 4 shows a plot representing an undesired power oscillation as a function of time;

[0042] Fig. 5 shows a schematic view of steps for determining an active power reference signal for compensation of an undesired power oscillation according to example embodiments; and

[0043] Fig. 6 shows a schematic view of steps for determining a reactive power reference signal for compensation of an undesired power oscillation according to example embodiments.Detailed Description of Embodiment(s)

[0044] The present disclosure relates to mitigation of undesired power oscillations propagating in a power system or power grid. Figure 1 shows an example topology of such a power grid 1 . Power grid 1 may, for example, be a high-voltage-direct- current, HVDC, system, a flexible alternating current transmission system, FACTS, a battery energy storage system, BESS, a photovoltaic, PV, power plant, or a wind power plant. Power grid 1 may, for example, be a smart grid.

[0045] The grid 1 comprises grid-connected nodes 10, 12, 14, 16 that are interconnected by transmission lines 17. Node 10 represents a phasor measurement unit, PMU, e.g. a micro phasor measurement unit, pPMU, indicated by a circle filled with a lightest shade of grey. Node 14 represents a power conversion station, PCS, indicated by a black circle. Node 16 comprises both a PMU and a PCS, and is indicated by a half-light, half-black filled circle. A PMU may also be referred to as a power control unit. PMU node 10 is a so-called grid-following or grid-feeding power converter. Nodes 14, 16 are so-called grid-forming power converters comprising power electronics components. Nodes 10, 14 and 16 are in communication with a controller circuitry 100 configured to perform power oscillation compensation. Node 12 is a node that is not in communication with controller 100 and is indicated by a circle in an intermediate shade of grey. The controller 100 is configured to receive phasor angle measurements from the PMUs and PCSs and is configured to provide time delays to the PCSs. The PCSs are configured to apply the time delays during power injection into the grid, and / or during power absorption from the grid, so as to compensate for an undesired power oscillation. In this example, the PMUs and PCSs are power measurement systems, PMSs, and the PCSs are also power compensating units, PCUs.

[0046] Subplot 101 shows an electrical quantity measured at node 12. Subplot 102 shows the same electrical quantity measured at node 14. Subplot 103 shows the electrical quantity measured at node 16. In this example, the electrical quantity is the voltage measured at the respective nodes. Subplots 101 , 102, 103 have the same y-axis 180 representing the voltage, and have the same x-axis 170 representing time. The corresponding graphs 110, 120, 130 are sinusoidal waves over a single period that are shifted in time with respect to each other. The plots are measured over the same time interval and reveal an out of phase behaviourbetween the nodes 12, 14 and 16. It is noted that nodes 12, 14 are located at the ends of outer regions 150, 160 of the grid 1 respectively, while node 16 is located near the geographical centre of the grid 1. Figure 1 shows that there is a 180 degrees phase shift, corresponding to a phase shift of half a period, between the voltage at node 12 and the voltage at node 14; and that there is a 90 degrees phase shift, corresponding to a phase shift of a quarter of a period, between the voltage at centre node 16 with respect to the outer nodes 12, 14. To obtain measurements and plots such as 101 , 102, 103, PMlls may timestamp the measurements by use of global positioning system, GPS, signals. This allows determining the measurements at different locations at a same moment in time.

[0047] The power system 1 has buses, e.g. slack buses, generation buses and load buses, across which the power oscillations occur. The undesired power oscillations are low-frequency oscillations and may occur at a few hundred millihertz, mHz, e.g. 0.1 Hz to 1 Hz, in different power system areas 150, 160. To compensate for undesired power oscillations, the following steps may be taken: a) real-time oscillation characterisation, i.e. collecting measurements from the PMlls 10, 16 to determine oscillation parameters or, in other words, the system matrix; b) phase shift and delay analysis, i.e. determining compensating time delays based on the system matrix; c) setting active and reactive power references, i.e. applying the time delays to damp the undesired oscillation; and d) adaptive feedback, i.e. continuously performing the steps a)-b)-c), thereby updating the system matrix and time delays over time.

[0048] Figure 2 illustrates steps 201 , 202, 203, 204, 205, 206 of a method for compensating the undesired power oscillations propagating throughout power grid 1. In a first step, an N-by-M matrix, 210 comprising voltage phasor angle measurements 9, is obtained from the power measurement systems in power system 1 . N is the number of PMSs and M is the number of time instances at which the voltage phasor angle is regarded.

[0049] Next, in step 201 , dynamics among the voltage phasor angles are linearly approximated, resulting in system matrix A, 220. The system matrix 220 provides a system model relating to the power grid’s inertia distribution and energy flow. Thesystem matrix 220 holds oscillatory characteristics derived from the voltage phasor angles 9.

[0050] In step 202, an eigenvalue decomposition of the system matrix 220 is determined. The result of step 202 is an eigenvector matrix 221 , comprising the eigenvectors of matrix A as its columns, and an eigenvalue matrix 222, comprising the eigenvalues of matrix A on its diagonal. In the general case, the eigenvectors and eigenvalues of system matrix A, 220, are complex. In other words, entries of the eigenvectors comprise a real part and an imaginary part and the eigenvalues comprise a real part and an imaginary part. Each eigenvector-eigenvalue pair relates to a single power oscillation. The real part of an eigenvalue represents a damping factor, while the imaginary part of an eigenvalue represents an angular frequency of the corresponding oscillation. The components, or, in other words, vector entries, of an eigenvector indicate the contributions to the corresponding oscillation at the physical locations of the PMSs respectively. In other words, each eigenvector characterises a single oscillation component, wherein an entry of the eigenvector represents a contribution to the corresponding oscillation component at one of the PMSs.

[0051] After determining the eigenvalues, one of the eigenvalues is selected as the so-called dominant eigenvalue 22 in step 203. The dominant eigenvalue 22 corresponds to the undesired power oscillation that will be compensated for. In this example, the eigenvalue having the largest real part is chosen, thereby targeting the least damped power oscillation that was captured in the system matrix 220. By deciding on the dominant eigenvalue 22, the corresponding dominant angular frequency 280, i.e. the imaginary part of the dominant eigenvalue, and corresponding dominant eigenvector 21 are automatically fixed.

[0052] Further, in step 204, two furthest entries 231 , 232 in the dominant eigenvector 21 that have a largest difference of arguments among all entries in the dominant eigenvector 21 are determined. This step is further illustrated in subplot 301 of Figure 3. Subplot 301 shows the imaginary plane with real axis 241 and imaginary axis 242. The complex entries of eigenvector 21 are plotted in the imaginary plane. It can be observed that in this example, two regions 235, 236 are formed whereentries occur in a group. Each entry of vector 21 corresponds to a contribution of a single PMS of the grid 1 to the dominant oscillation. Therefore, the locations of the entries in the imaginary plane can be related to geographical locations within the power grid 1. As such, the regions 235, 236 represent PMSs in the grid 1 that oscillate against each other and correspond to physical regions that are located at outer ends of the grid 1 , such as for example regions 150, 160 illustrated in Figure 1.

[0053] The furthest entries 231 , 232 are the entries forming the largest angle 300, in absolute value, between them in the complex plane. The largest possible angle is 180 degrees, since argument differences that are larger than 180 degrees have a negative equivalent that is smaller in absolute value. As an example, the argument of a first entry of vector 21 may be 0 degrees, and the argument of a second entry of vector 21 may be 240 degrees. The difference of these arguments is regarded as -120 degrees or 120 degrees and not as 240 degrees, since it indicates the proximity over a rotation within the complex plane. In other words, the furthest entries 231 , 232 are those entries that, in the complex plane, form an angle between them that is closest to 180 degrees, among all combinations of entries of the vector 21. Coincidentally, in subplot 301 of Figure 3, the furthest entries 231 , 232 also coincide with the entries located furthest away from each other in the complex plane. It is noted that not the distance within the complex plane is what defines entries 231 , 232 as the furthest entries, but the angle formed between them, which approximates 180 degrees.

[0054] In the next step 205, all entries of the dominant eigenvector are projected in the complex plane along directions defined by the furthest entries. The two directions in the complex plane indicated by the two furthest entries 231 , 232 that are a rotation closest to 180 degrees away from each other, are being inserted into the columns of a 2-by-2 matrix. The inverse of this 2-by-2 matrix is then used as a projection matrix to project all the entries of the vector 21. The inverse may, for example, be the Moore-Penrose inverse. As such, the assumption that a 180 degree phase shift exists within outer regions of the grid 1 is imposed on the entries of the eigenvector 21 by the projection. Subplot 302 of Figure 3 illustrates the projection executed in step 205. In subplot 302, the complex plane is illustrated in terms ofarguments 310 and magnitudes 311. Before the projection, entries 30, 31 , 32, 33, 34 of the eigenvector 21 may have arguments very close to each other. After the projection, the projected entries corresponding to the entries 30, 31 , 32, 33, 34 become spread out, having arguments between zero and 180 degrees. The projected entries make up a projected eigenvector 25. By the projection, an entry is shifted within the complex plane in such a way that its new argument indicates a voltage component resulting from the undesired power oscillation as a phasor. In particular, it concerns the voltage component that occurs at the PMU corresponding to the entry as a result of the dominant power oscillation. As such, the compensating phase angles may be retrieved as the arguments of the projected entries in step 206.

[0055] After step 206, time delays 270 are determined for one or more power compensating units 12, PCUs, in the power grid 1. The compensating phase angles 260 are related to the time delays 270 by the angular frequency 280. By applying the time delays 270 in the corresponding PCUs, a compensation of the undesired power oscillation at the dominant angular frequency can be achieved.

[0056] Figure 4 illustrates the effect of compensating a power oscillation. The y-axis 180 represents an electrical metric, in this case the measured voltage amplitude. The x-axis indicates time. The oscillation at a certain angular frequency, without application of the method, is indicated by reference numeral 401 . Without active compensation, oscillation damping may take tens of seconds. The measured voltage after compensating the undesired power oscillation is indicated by reference numeral 402 and is already dampened after a few oscillations.

[0057] Turning to Figure 5, a schematic is shown that indicates how the time delays 270 may be further processed to perform step c) by setting an active power reference at a particular PCU. Setting the active power reference may be based on the time derivative of the voltage phasor angle measurements 503 that are measured at the PCU. The derivative 503 may be taken as the input, i.e. as the active power steering signal. In a first block 510, a band-pass filtering operation may be performed, using the dominant angular frequency 280 as the frequency to allow through. As a result, the filtered active power steering signal indicates the presenceof the dominant angular frequency 280 in the PCU. The magnitude of the result determines how large the power reference will be set for the compensation. Next, the time delay 270 found for the PCU is applied in block 520. As an example, assume the compensating phase angle for the PCU is 35 degrees, or 0.61 radians, and the dominant oscillation frequency is 0.5 Hz, corresponding to a dominant angular frequency of 3.14 rad / s. The corresponding time delay is 0.194 seconds. Consequently, in the delay block 520, a transport delay of e-0 194sor 1 / e°194smay be applied. Finally, the resulting outcome is passed through block 530, where an active gain Kpmay be applied to provide a normalised active power reference P, 535.

[0058] Figure 6 shows a schematic that indicates how the time delays 270 may be further processed to perform step c) by setting a reactive power reference. The approach is similar to setting the active power reference. The reactive power reference setting may take into account the time derivative of the power transfer 504. The reactive power steering signal may be set equal to the derivative 504. In block 510 a band-pass filtering is applied using the dominant angular frequency 280 as a centre frequency, as for the active power reference setting. For reactive power, a 90 degree offset is applied to the time delay before delaying the filtered reactive power steering signal in block 520. A reactive gain Kq may be applied to provide a normalised reactive power reference Q, 545. The setting of the active and reactive power references may be performed simultaneously in every PCU of the power grid 1.

[0059] As used in this application, the term “circuitry” may refer to one or more or all of the following: (a) hardware-only circuit implementations (such as implementations in only analog and / or digital circuitry) and (b) combinations of hardware circuits and software, such as (as applicable): (i) a combination of analog and / or digital hardware circuit(s) with software / firmware and (ii) any portions of hardware processor(s) with software (including digital signal processor(s)), software, and memory(ies) that work together to cause an apparatus, such as a mobile phone or server, to perform various functions) and (c) hardware circuit(s) and or processor(s), such as a microprocessor(s) or a portion of a microprocessor(s), that requires software (e.g., firmware) for operation, but the software may not be present when it is not needed for operation. This definition of circuitry applies to all uses of this term in thisapplication, including in any claims. As a further example, as used in this application, the term circuitry also covers an implementation of merely a hardware circuit or processor (or multiple processors) or portion of a hardware circuit or processor and its (or their) accompanying software and / or firmware. The term circuitry also covers, for example and if applicable to the particular claim element, a baseband integrated circuit or processor integrated circuit for a mobile device or a similar integrated circuit in server, a cellular network device, or other computing or network device.

[0060] Although the present invention has been illustrated by reference to specific embodiments, it will be apparent to those skilled in the art that the invention is not limited to the details of the foregoing illustrative embodiments, and that the present invention may be embodied with various changes and modifications without departing from the scope thereof. The present embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. In other words, it is contemplated to cover any and all modifications, variations or equivalents that fall within the scope of the basic underlying principles and whose essential attributes are claimed in this patent application. It will furthermore be understood by the reader of this patent application that the words "comprising" or "comprise" do not exclude other elements or steps, that the words "a" or "an" do not exclude a plurality, and that a single element, such as a computer system, a processor, or another integrated unit may fulfil the functions of several means recited in the claims. Any reference signs in the claims shall not be construed as limiting the respective claims concerned. The terms "first", "second", third", "a", "b", "c", and the like, when used in the description or in the claims are introduced to distinguish between similar elements or steps and are not necessarily describing a sequential or chronological order. Similarly, the terms "top", "bottom", "over", "under", and the like are introduced for descriptive purposes and not necessarily to denote relative positions. It is to be understood that the terms so used are interchangeable under appropriate circumstances and embodiments of the invention are capable of operatingaccording to the present invention in other sequences, or in orientations different from the one(s) described or illustrated above.

Claims

CLAIMS1 . Method for compensating for undesired power oscillations (110; 120) propagating throughout a power grid (1 ), the method comprising the steps of:- obtaining, from N power measurement systems (10), PMSs, in the power grid, phasor angle measurements (210) of an electrical metric characterising said undesired power oscillations at M consecutive time instances;- determining (201 ), based on the measurements, a linear approximation of dynamics among the phasor angles of the N PMSs, thereby obtaining a system matrix (220);- performing (202) an eigenvalue decomposition of the system matrix, thereby obtaining N complex eigenvectors (221 ) and N complex eigenvalues (222);- selecting (203) a dominant eigenvalue (22) associated to an undesired power oscillation at an angular frequency (280) and a corresponding dominant eigenvector (21 );- identifying (204) two furthest entries (231 ;232) in the dominant eigenvector having a largest difference between arguments among all entries in the dominant eigenvector;- projecting (205) entries of the dominant eigenvector in a complex plane along directions defined by the furthest entries, resulting in a projected eigenvector (25) having N complex projected entries, characterising compensating phase angles (260) of the undesired power oscillation in the N PMSs;- determining (206), for one or more power compensating units (12), PCUs, in the power grid, one or more respective time delays (270) from the N compensating phase angles for compensation of the undesired power oscillation at the angular frequency.

2. The method according to claim 1 , wherein the determining of the system matrix comprises performing a dynamic mode decomposition, DMD, algorithm.

3. The method according to claim 1 or 2, wherein the selecting of the dominant eigenvalue comprises selecting the eigenvalue corresponding to a most critical undesired power oscillation.

4. The method according to claim 3, wherein the selecting of the dominant eigenvalue comprises selecting the eigenvalue having the smallest absolute value of its real part.

5. The method according to claim 1 or 2, wherein the selecting of the dominant eigenvalue comprises selecting the eigenvalue having an imaginary part closest to a predetermined angular frequency, thereby targeting a specific undesired power oscillation corresponding to the predetermined angular frequency.

6. The method according to any one of the preceding claims, wherein the determining of the time delays comprises scaling the projected entries with an inverse of the angular frequency.

7. The method according to any one of the preceding claims, wherein the determining of a time delay in a PCU and / or PMS comprises interpolating compensating phases in nearby PMSs based on a distance to the nearby PMSs.

8. The method according to any one of the preceding claims, further comprising, by at least one of the PCUs:- determining an active power steering signal (503) based on a derivative of phasor angle measurements in the at least one PCU;- applying a band pass filter, having a centre frequency equal to the angular frequency of the undesired power oscillation, to the active power steering signal;- applying the time delays for the at least one PCU to the filtered active power steering signal; and- performing active power injection based on the delayed active power steering signal.

9. The method according to any one of the preceding claims, further comprising, by at least one of the PCUs:- determining a reactive power steering signal (504) based on a derivative of power transfer in the at least one PCU;- applying a band pass filter, having a centre frequency equal to the angular frequency of the undesired power oscillation, to the reactive power steering signal;- applying the time delays for the at least one PCU, offset by 90 degrees, to the filtered reactive power steering signal; and - performing reactive power injection based on the delayed reactive power steering signal.

10. A data processing system comprising means for carrying out the method of any one of the preceding claims.11 . A computer program comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method according to any one of the claims 1 -9.

12. A computer-readable medium comprising instructions which, when executed by a computer, cause the computer to carry out the method according to any one of the claims 1-9.

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