THÉVENIN EQUIVALENT BASED STATIC CONTINGENCY ASSESSMENT

Abstract

The present invention relates to a method for static security assessment of a power system and a real time static security assessment system for assessing a power system, the power system having a plurality of generators, the plurality of generators being represented in the network by a plurality of voltage controlled nodes, wherein the method for static security assessment of the power system comprises receiving information of a present state of the power system, determining a Thevenin equivalent for each voltage controlled node, determining for each voltage controlled node on basis of the determined present state of the power system and determining a first representation of the network based on the determined Thevenin equivalents, determining a modified representation of the network, wherein the modified representation is a representation of the network having at least one contingency, wherein at least one Thevenin equivalent of at least one voltage controlled node is modified due to the at least one contingency, the modified network representation being determined on the basis of the modified Thevenin equivalents, calculating voltage angles of the modified Thevenin equivalents, and evaluating the voltage angles to determine whether the network having at least one contingency admit a steady state. Also a method of providing information on a real time static security assessment of a power system is disclosed.

Claims

1. A method for conducting contingency analyses in static security assessment of a power system, the power system having a plurality of generators injecting power into a network having a plurality of nodes and a plurality of branches, the plurality of generators being represented in the network by a plurality of voltage controlled nodes, the method comprising: receiving information of a present state of the power system, determining a Thvenin equivalent for each voltage controlled node, wherein a Thvenin equivalent is determined for each voltage controlled node on the basis of the determined present state of the power system, determining a representation of the network based on the determined Thvenin equivalents, applying at least one contingency to the network, determining a modified representation of the network, wherein the modified network representation is a representation of the network having at least one applied contingency, wherein at least one Thvenin equivalent of at least one voltage controlled node is modified due to the at least one applied contingency, the modified network representation being determined on the basis of the modified Thvenin equivalents, calculating voltage angles of the modified Thvenin equivalents, and evaluating the voltage angles to determine whether the network having at least one applied contingency admits a steady state.

2-17. (canceled)

18. The method according to claim 1, wherein the representation of the network is based on a two-source equivalent, wherein the two-source equivalent comprises the determined Thvenin equivalent and a voltage phasor of the voltage controlled node.

19. The method according to claim 1, wherein the method comprises evaluating whether the application of the contingency results in a stable network condition.

20. The method according to claim 1, wherein the at least one contingency is a topological change to the network.

21. The method according to claim 1, wherein the at least one contingency is a broken transmission line grid, a loss of a single transmission line, a loss of a generator, a damaged generator and/or any fault that provide a fault to the power system that may result in an unstable power system.

22. The method according to claim 1, wherein voltages at voltage controlled nodes and/or at non-controlled voltage nodes are compared against operational limits.

23. The method according to claim 1, wherein the calculation of the Thvenin equivalent for each voltage controlled node is performed assuming a constant active power injection and constant voltage magnitudes for each voltage controlled node.

24. The method according to claim 1, wherein a grid transformation matrix comprises calculated Thvenin voltages for each voltage controlled node, one or more corresponding grid transformation coefficients and one or more corresponding voltages of voltage controlled nodes and/or wherein a grid transformation coefficient is a relation between the Thvenin equivalent voltage at a voltage controlled node and voltage phasors at neighbouring voltage controlled nodes.

25. The method according to claim 22, wherein the determined Thvenin equivalents on which the first network representation is based corresponds to Thvenin equivalents on which the modified network representation is based in at least the part of the modified network representation corresponding to a part of the first network representation.

26. The method according to claim 1, wherein the step of determining a present state of the power system comprises obtaining synchronized Phasor Measurement Unit measurements from a plurality of nodes of the power system.

27. The method according to claim 1, wherein the Thvenin equivalents, the modified Thvenin equivalents and/or the voltage angles are determined in real-time.

28. The method according to claim 1, wherein the Thvenin equivalent comprises a Thvenin voltage and a Thvenin impedance, and wherein determined Thvenin voltages are re-calculated based on the calculated voltage angles of the modified Thvenin equivalents, and modified voltage angles are calculated on basis of the updated Thvenin voltages and wherein a change in voltage angle is evaluated.

29. The method according to claim 28, wherein evaluating the change in voltage angle is performed until a convergence criterion is satisfied.

30. A computer program comprising a program code configured to perform the method according to claim 1, when executed on a computer.

31. A computer readable medium having stored thereon a program code configured to perform the method according to claim 1, when executed on a computer.

32. A real time static security assessment system for conducting contingency analyses in a power system, the power system having a plurality of generators injecting power into a network having a plurality of nodes and a plurality of branches, the plurality of generators being represented in the network by a plurality of nodes of power injection, the system comprises: a data processor configured to: receive information of a present state of the power system, determine a Thvenin equivalent for each voltage controlled node, wherein a Thvenin equivalent is determined for each voltage controlled node on the basis of the determined present state of the power system, determine a representation of the network based on the determined Thvenin equivalents, apply at least one contingency to the network, determine a modified representation of the network, wherein the modified network representation is a representation of the network having at least one applied contingency, wherein at least one Thvenin equivalent of at least one voltage controlled node is modified due to the at least one contingency, the modified network representation being determined on the basis of the modified Thvenin equivalents, calculating voltage angles of the modified Thvenin equivalents, evaluating the voltage angles to determine whether the network having at least one applied contingency admits a steady state, and an interface configured to output information on the static security assessment of the modified network representation of the network, wherein the information comprises the evaluated voltage angle.

33. A method of providing information on a real time static security assessment of a power system, the power system having a plurality of generators injecting power into a network having a plurality of nodes and a plurality of branches, the plurality of generators being represented in the network by a plurality of nodes of power injection, the method comprising: receiving information of a present state of the power system, determining a two source Thvenin equivalent representation, where the representation includes the power system as seen from each voltage controlled node, wherein a Thvenin equivalent is determined for each voltage controlled node on basis of the determined present state of the power system, wherein the Thvenin equivalent comprises a Thvenin voltage and a Thvenin impedance, determining a representation of the network based on the determined Thvenin equivalents, applying at least one contingency to the network, determining a modified representation of the network representation, wherein the modified network representation of the network having at least one applied contingency, wherein at least one Thvenin equivalent of at least one voltage controlled node is modified due to the at least one contingency, the modified network representation being determined on the basis of the modified Thvenin equivalents, calculating voltage angles of the modified Thvenin equivalents, evaluating the voltage angle to determine whether the network having at least one applied contingency admits a steady state, and outputting information on static security assessment of the modified representation of the network, wherein the information comprises the evaluated voltage angles.

Description

BRIEF DESCRIPTION OF THE DRAWING

[0051] FIGS. 1a-c shows an overview of a power system and corresponding measurements; FIG. 1a shows an electric power system, FIG. 1b shows synchronized measurements from two nodes of the electric power system, and FIG. 1c shows the resulting phasors in an impedance plane,

[0052] FIG. 2 shows a generalized electric power system, where system loads are represented as impedances and the generators are assumed to maintain constant terminal voltage,

[0053] FIG. 3 is a flow chart of a method according to the present invention,

[0054] FIG. 4 illustrates a two source Thvenin equivalent representation,

[0055] FIG. 5 shows an active power balance for a synchronous generator,

[0056] FIGS. 6a-b show schematically power networks comprising a plurality of voltage controlled nodes and non-controlled nodes,

[0057] FIG. 7a shows a representation of a network having coupled two source Thvenin equivalent representation, and FIG. 7b shows a grid transformation matrix obtained from the representation of the network in FIG. 7a,

[0058] FIG. 8 is a flow chart of a method of providing information on a real time static security assessment of a power system.

[0059] FIG. 9 is a flow chart illustrating a method of real time static security assessment,

[0060] FIG. 10 shows a simulation result of a method according to the present invention and of Newton Raphson's power flow method,

[0061] FIG. 11 shows simulation results of a further embodiment of the method according to the present invention,

[0062] FIGS. 12A and 12B show a Nordic 32 test system and simulation results of a method according to the present invention, respectively.

DETAILED DESCRIPTION OF THE DRAWING

[0063] The present invention will now be described more fully hereinafter with reference to the accompanying drawings, in which exemplary embodiments of the invention are shown. The invention may, however, be embodied in different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. Like reference numerals refer to like elements throughout. Like elements will, thus, not be described in detail with respect to the description of each figure.

[0064] In the present description the term secondary network and a part of a network may in the following be used to indicate a part of the network being evaluated isolated from the rest of the network.

[0065] FIG. 1a shows a power system 1, where a Phasor Measurement Unit (PMU), or another measurement device that provide synchronized measurements in real time, of voltage and current phasors along with frequency measurements, is installed at node 1 and node 2. The synchronized measurements are shown in FIG. 1b, for node 1 and node 2, respectively.

[0066] FIG. 1c shows the resulting phasors and plotted in the same complex plane. The phase difference between the signals from node 1 and node 2, respectively, is indicated.

[0067] An exemplary power system 10 is shown in FIG. 2. FIG. 2 shows the power system 10 where all loads are represented as constant impedances 13 and where all generators 11 are assumed to maintain a constant terminal voltage. With all system impedances 13 known, the system operating conditions can be determined from the generators 11 terminal voltages. The power system 10 comprises the generators 11 and the network 14. In the network 14, the generators are represented by a plurality of voltage controlled nodes, or nodes of power injection, 16. Non-controlled nodes 15 and the impedances 13 are interconnected via branches 12. The generators are in FIG. 2 assumed to maintain a constant terminal voltage. In the following this generalized notation will be referred to when discussing the network further.

[0068] The Thvenin impedance seen from a given voltage controlled node is the impedance which can be measured if all other voltage controlled nodes were to be short circuited.

[0069] FIG. 3 is a flow chart of a method 1 for static security assessment of a power system 10, such as for a contingency analysis in a static security assessment of a power system. The power system having a plurality of generators 11 injecting power {right arrow over (S)}.sub.j into a network 14 having a plurality of nodes (15, 16) and a plurality of branches 12. The plurality of generators 11 are represented in the network 14 by a plurality of voltage controlled nodes 16.

[0070] In step 1a, information of a present state of the power system is received, and in step 1b, a Thvenin equivalent for each voltage controlled node 16 is determined, wherein a Thvenin equivalent is determined for each voltage controlled node 16 on basis of the determined present state of the power system 10.

[0071] In step 1c, a first representation of the network 14 based on the determined Thvenin equivalents is determined, and in step 1d, a modified representation of the network 14 is determined, wherein the modified representation is a representation of the network 14 having at least one contingency, wherein at least one Thvenin equivalent of at least one voltage controlled node 16 is modified due to the at least one contingency. The modified network representation may be determined on the basis of the modified Thvenin equivalents.

[0072] In step 1e, voltage angles .sub.j0 of the modified Thvenin equivalents are calculated, and in step 1f, the voltage angles .sub.j0 are evaluated to determine whether the network 14 having at least one contingency is in steady state. The evaluation may be performed using any PMU based evaluation methods. The voltage angels .sub.j0 of the modified Thvenin equivalents may be calculated for voltage controlled nodes and/or for non-controlled nodes.

[0073] The method may optionally comprise the step 1g, in which synchronized Phasor Measurement Unit measurements are initially obtained from a plurality of nodes 15, 16 of the power system 10.

[0074] In an exemplary method, at least one contingency may be applied to the network 14 in step 1c, before the voltage angles .sub.j0 are evaluated in step 1f to determine whether the application of the contingency results in a stable network condition, thus to evaluate whether the network admits a steady state.

[0075] The method may comprise the optional steps 1h and 1i.

[0076] Thus, in a further exemplary method, in step 1h, a change in voltage angle is evaluated by comparing a recalculated modified voltage angle .sub.j1 to the calculated voltage angle .sub.j0, and wherein the step of recalculation is repeated until the change in voltage angle fulfils a convergence criterion, for example until the change in voltage angle is below a predetermined voltage angle change threshold.

[0077] In another exemplary method, the voltages at non-controlled nodes 15 may be obtained, for example using a linear model.

[0078] In step 1i, the resulting post-contingency voltages may be evaluated compared against operational limits of the network or power system.

[0079] Additionally, in an exemplary method, in 1f, the Thvenin equivalent comprises a Thvenin voltage {tilde over (E)}.sub.th and a Thvenin impedance Z.sub.th, and wherein determined Thvenin voltages {tilde over (E)}.sub.th are re-calculated based on the calculated voltage angles .sub.j0, of the modified Thvenin equivalents, and re-calculated modified voltage angles .sub.j1, are calculated on basis of the re-calculated Thvenin voltages and wherein a change in voltage angle is evaluated.

[0080] The method is provided to enable a static security assessment of the system, such as to conduct a contingency analysis in static security assessment of the system. It is known that when a contingency is applied to a power system, typically, a transient behaviour will be seen, and thus, the voltage angles may in some embodiments be evaluated when these transients have faded out and the power system is, or is assumed to be, in a static mode.

[0081] To determine when the power system is in a static mode an iterative re-calculation of voltage angles is subject to a convergence criterion. Such convergence criterion can be based on the size of change of voltage angles from one iteration to the next.

[0082] Thus, after the application of the contingency, the voltage angles are calculated based on the modified Thvenin equivalents, and in an iterative process, the Thvenin equivalents are re-calculated based on the calculated voltage angles, and re-calculated voltage angles are calculated based on the re-calculated Thvenin equivalents. The re-calculated voltage angles are compared with the calculated voltage angles, to provide a change in voltage angle, and subject to the convergence criterion, such when for example the change in voltage angle becomes lower than a threshold change in voltage angle, the power system is in a static mode, and an evaluation of the power system may be performed.

[0083] In another exemplary method, in 1f, the power flow in the network 14 is evaluated.

[0084] FIG. 4 shows a two-source Thvenin equivalent representation 17 of a power system 10 seen from a voltage controlled node N.sub.j. The two source Thvenin equivalent representation 17 comprises a voltage phasor {tilde over (V)}.sub.j and a Thvenin equivalent represented by the Thvenin voltage {tilde over (E)}.sub.th,j and the Thvenin impedance Z.sub.th,j, wherein the voltage phasor {tilde over (V)}.sub.j is the voltage phasor at the voltage controlled node N.sub.j and the Thvenin equivalent is representing the network as seen from the voltage controlled node N.sub.j.

[0085] The voltage phasor {tilde over (V)}.sub.j is given by a voltage magnitude |V.sub.j| and a voltage angle .sub.j determined at the voltage controlled node N.sub.j, i.e. {tilde over (V)}.sub.j=|V.sub.j|.sub.j.

[0086] The Thvenin equivalent comprises a Thvenin voltage {tilde over (E)}.sub.th,j and a Thvenin impedance Z.sub.th,j, wherein the Thvenin voltage is given by a Thvenin voltage magnitude |E.sub.th,j| and a Thvenin voltage angle .sub.th,j i.e. {tilde over (E)}.sub.th,j=|E.sub.th,j|.sub.th,j. The Thvenin impedance Z.sub.th,j is given by a Thvenin impedance magnitude |Z.sub.th,j| and a Thvenin impedance angle .sub.th,j, i.e. Z.sub.th,j=|Z.sub.th,j|.sub.th,j or Z.sub.th,j=R.sub.th,j+iX.sub.th,j.

[0087] The active power injection P.sub.j=Re{{right arrow over (S)}.sub.j} at the voltage controlled node N.sub.j is given by:

[00001]$P_j=\frac{X_{\mathrm{th},j}.Math..Math.V_j.Math..Math..Math.E_{\mathrm{th},j}.Math.}{R_{\mathrm{th},j}^2+X_{\mathrm{th},j}^2}.Math.\sin .Math..Math.{}_j-\frac{R_{\mathrm{th},j}.Math..Math.V_j.Math..Math..Math.E_{\mathrm{th},j}.Math.}{R_{\mathrm{th},j}^2+X_{\mathrm{th},j}^2}.Math.\cos .Math..Math.{}_j+\frac{R_{\mathrm{th},j}.Math.{.Math.V_j.Math.}^2}{R_{\mathrm{th},j}^2+X_{\mathrm{th},j}^2}$

[0088] FIG. 5 shows the power injection P.sub.j as a function of a voltage angle .sub.j at a voltage controlled node N.sub.j, when the voltage magnitude |V.sub.j| and the Thvenin equivalent are constants, and the P curve thus shows a relation between active power injection and voltage angle at a point of constant voltage and the system thus admits a steady state. Furthermore, a mechanical power P.sub.m of a rotor shaft configured to a generator is shown.

[0089] A voltage angle .sub.j of a voltage controlled node N.sub.j may be determined when the network 14 is in steady state, i.e. when the power injection P.sub.j from the voltage controlled node N.sub.j is at least equal to the mechanical power P.sub.m.

[0090] .sub.j,i represents an initial voltage angle as measured in a pre-fault operational mode. Thus, the initial or pre-fault voltage angle may be described by the curve 51 (the electrical output of the node), and the point of operation for the j'th node of the power system, N.sub.j, in the pre-fault condition, or pre-contingency condition, is illustrated by the intersection 55, wherein, in the steady state mode, the mechanical power P.sub.m equals the active power injection, P.sub.j. The post-fault, or post-contingency, condition, the point of operation .sub.j1 may be derived from the change in Thvenin equivalent, thus the voltage angle .sub.j1 may be calculated based on the modified Thvenin equivalents and the voltage angle may be described by the curve 53. The post-fault, or post-contingency, point of operation for the jth voltage controlled node, is illustrated by the intersection 57, wherein, in the steady state mode, the mechanical power equals the active power injection.

[0091] .sub.j0 may thus represent a calculated or measured voltage angle determined at the voltage controlled node N.sub.j before a contingency is applied to the network 14, and .sub.j1 may represent a modified voltage angle determined at the voltage controlled node N.sub.j after the contingency is applied to the network 14.

[0092] An unstable condition may occur if the power injection P.sub.j at the voltage controlled node N.sub.j does not exceed the mechanical input power P.sub.m. For example, the unstable condition may occur because of a broken transmission line grid or a broken generator.

[0093] The network may be represented by a plurality of the voltage controlled nodes, a plurality of non-controlled nodes, or voltages at nodes without voltage control, interconnected via branches. For the present analysis, it has proven advantageous to pre-suppose that each voltage controlled node is primarily influenced by neighbouring voltage controlled nodes, such as by first degree neighbouring voltage controlled nodes, or second degree neighbouring voltage controlled nodes. Hereby, for the analysis, as seen in FIG. 6a, a secondary network 14A may be configured in the network 14 and forming part of the network 14, for each voltage controlled node.

[0094] It is an advantage of representing the network using a secondary network for each voltage controlled node in that the analysis may then be performed using parallel computing.

[0095] The secondary network 14A is represented by a voltage controlled node N.sub.j looking into a plurality of other voltage controlled nodes (N.sub.X1-N.sub.X5)and multiple non-controlled nodes. Each voltage controlled node in the secondary network being illustrated by a solid black square and each non-controlled node in the secondary network being represented by a solid black circle. The multiple non-controlled nodes may for example be loads and may consume the power generated by the voltage controlled nodes (N.sub.x, N.sub.j).

[0096] The voltage controlled nodes and the non-controlled nodes outside of the secondary network, and thus not forming part of the secondary network are illustrated by white squares and white circles, respectively.

[0097] FIG. 6a shows the secondary network 14A and in this particular example and for the purpose of determining the Thvenin equivalent of the network as seen from the N.sub.j node, the secondary network 14A is represented by a voltage controlled node N.sub.j, a plurality of short circuited voltage controlled nodes (N.sub.X1-N.sub.X5) and multiple non-controlled nodes.

[0098] FIG. 6b shows a secondary network 14A, and corresponding Thvenin equivalents and grid transformation coefficients.

[0099] An open-circuit is established at the voltage controlled node N.sub.j, and the Thvenin voltage {tilde over (E)}.sub.th,j may be determined as seen from the voltage controlled node N.sub.j. The grid transformation coefficients may be determined from the network in that the secondary network 14A comprises another voltage controlled node N.sub.k. A unit current is injected at the other voltage controlled node N.sub.k while short circuiting all remaining voltage controlled nodes (N.sub.x1-N.sub.x5). With the configuration of the secondary network 14A, a grid transformation coefficient k.sub.jk may be determined as a relation between a voltage phasor {tilde over (V)}.sub.j determined at the voltage controlled node N.sub.j and at least a voltage phasor {tilde over (V)}.sub.k determined at another voltage controlled node N.sub.k.

[0100] Alternatively, the grid transformation coefficient may be defined as a relation between the Thvenin equivalent voltage {tilde over (E)}.sub.th,j determined at the voltage controlled node N.sub.j and the voltage phasors determined at any other voltage controlled nodes.

[0101] As a further alternative, the grid transformation coefficient may be defined as a relation between the Thvenin equivalent voltage {tilde over (E)}.sub.th,j calculated at the voltage controlled node N.sub.j and voltage phasors determined at neighbouring voltage controlled nodes.

[0102] FIG. 7a shows a representation of the network having four voltage controlled nodes, each being expressed by coupled two-source Thvenin equivalents. The representation of the network 14 is based on a two-source equivalent (17A-17D), wherein the two-source equivalent comprises the determined Thvenin equivalent and a corresponding voltage phasor of a corresponding voltage controlled node.

[0103] A first representation 17A of the network 14 is seen from a voltage controlled node N.sub.j, having a Thvenin equivalent of the voltage controlled node N.sub.j and having at least one other voltage controlled node N.sub.k with a voltage phasor {tilde over (V)}.sub.k. The relation between a Thvenin voltage {tilde over (E)}.sub.th,j and the voltage phasor {tilde over (V)}.sub.k of the other voltage controlled node is represented by grid transformation coefficient k.sub.jk.

[0104] A second representation 17B of the network 14 is seen from a voltage controlled node N.sub.k. Thvenin equivalent of the voltage controlled node N.sub.k representing a secondary network 14A having at least two other voltage controlled nodes (N.sub.j,N.sub.l), wherein a voltage controlled node N.sub.j, with a voltage phasor {tilde over (V)}.sub.j, and a voltage controlled node N.sub.l, with a voltage phasor {tilde over (V)}.sub.l, are each related to a Thvenin voltage {tilde over (E)}.sub.th,k of the voltage controlled node N.sub.k through grid transformation coefficients k.sub.kj and k.sub.kl, respectively.

[0105] A third representation 17C of the network 14 is seen from a voltage controlled node N.sub.l. Thvenin equivalent of the voltage controlled node N.sub.l representing a secondary network 14A having at least two other voltage controlled nodes (N.sub.k,N.sub.i), wherein a voltage controlled node N.sub.k, with a voltage phasor {tilde over (V)}.sub.k, and a voltage controlled node N.sub.i, with a voltage phasor {tilde over (V)}.sub.i, are each related to a Thvenin voltage {tilde over (E)}.sub.th,l of the voltage controlled node N.sub.l through grid transformation coefficients k.sub.lk and k.sub.li, respectively.

[0106] A fourth representation 17D of the network 14 is seen from a voltage controlled node N.sub.i, wherein Thvenin equivalent of the voltage controlled node N.sub.i represent a secondary network 14A having at least one other voltage controlled node N.sub.l with a voltage phasor {tilde over (V)}.sub.l. The relation between a Thvenin voltage {tilde over (E)}.sub.th,i and the voltage phasor {tilde over (V)}.sub.l of the other voltage controlled node N.sub.l is represented by a grid transformation coefficient k.sub.il.

[0107] FIG. 7b shows a grid transformation matrix 19 comprising the calculated Thvenin voltages ({tilde over (E)}.sub.th,j, {tilde over (E)}.sub.th,k, {tilde over (E)}.sub.th,l, {tilde over (E)}.sub.th,i) for each four voltage controlled nodes (N.sub.j, N.sub.k, N.sub.l, N.sub.i), one or more corresponding grid transformation coefficients and one or more corresponding voltage phasors ({tilde over (V)}.sub.j, {tilde over (V)}.sub.k, {tilde over (V)}.sub.l, {tilde over (V)}.sub.i)of the voltage controlled nodes (N.sub.j, N.sub.k, N.sub.l, N.sub.i).

[0108] It is an advantage of the grid transformation matrix 19 that it clearly indicates whether or not a direct coupling exists between two voltage controlled nodes in terms of a corresponding grid transformation coefficient.

[0109] FIG. 8 is a flow chart of a method 10 for providing information on a real time static security assessment of a power system 10. The power system 10 having a plurality of generators 11 injecting power {right arrow over (S)}.sub.j into a network 14 having a plurality of nodes (15, 16) and a plurality of branches 12. The plurality of generators 11 are represented in the network 14 by a plurality of voltage controlled nodes 16.

[0110] In step 10a, information of a present state of the power system 10 is received, and in step 10b, a Thvenin equivalent for each voltage controlled node 16 is determined, wherein a Thvenin equivalent may be determined for each voltage controlled node 16 on basis of the determined present state of the power system 10.

[0111] In step 10c, a first representation of the network 14 based on the determined Thvenin equivalents is determined, and in step 10d, a modified representation of the network 14 is determined, wherein the modified representation is a representation of the network 14 having at least one contingency, wherein at least one Thvenin equivalent of at least one voltage controlled node 16 is modified due to the at least one contingency. The modified network representation is determined on the basis of the modified Thvenin equivalents.

[0112] In step 10e, voltage angles .sub.j0 of the modified Thvenin equivalents are calculated, and in step 10f, the voltage angles .sub.j0 are evaluated to determine whether the network 14 having at least one contingency is in steady state.

[0113] In step 10g, the method is configured to output information comprising evaluated voltage angles on static security assessment of the modified representation of the network, wherein the information comprises the evaluated voltage angles.

[0114] The information may be output to a second system configured to determine a remedial control action for a power system 10 having a plurality of generators 11 that are in an unstable or insecure state, especially to real-time determination of remedial control actions to be carried out.

[0115] Furthermore, the information may be output to a third system configured to assessing stability of a power system 10 having a plurality of generators 11, especially to real-time stability assessment of the power system 10. Additionally, the third system may also relate to a determination of stability boundary conditions for the power system 10, and a determination of the system 10 security margins.

[0116] FIG. 9 is flow chart of a method 11 for conducting contingency analyses in static security assessment of a power system. The power system having a plurality of generators injecting power into a network having a plurality of nodes and a plurality of branches, the plurality of generators being represented in the network by a plurality voltage controlled nodes. The method comprises following steps: [0117] 11a) receiving information of a present state of the power system, [0118] 11b) determining a Thvenin equivalent for each voltage controlled node, such as a two source Thvenin equivalent, wherein the Thvenin voltages (V.sub.th) and Thvenin impedances (Z.sub.th) are calculated for each voltage controlled node on basis of the determined present state of the power system, [0119] 11c) determining a grid transformation matrix based on the calculated Thvenin equivalents (V.sub.th, Z.sub.th, and I.sub.th), and wherein the grid transformation matrix comprises the calculated Thvenin equivalents for each voltage controlled node, [0120] 11d) applying perturbations to the grid transformation matrix and thereby modifying the Thvenin equivalents with at least one contingency, [0121] 11e) calculating voltage angles of the modified Thvenin equivalents on basis of the two source Thvenin equivalents for each predetermined selection of voltage controlled nodes, and wherein the Thvenin voltages (V.sub.th) are updated with the calculated voltage angles, and new voltage angles are calculated on basis of the updated Thvenin voltages.

[0122] FIG. 10 shows a simulation result of a method according to the present invention and of Newton Raphson's power flow method (NR).

[0123] In this specific example, the method according to the present invention is denoted as Thvenin Equivalent based Static Contingency Assessment (TESCA).

[0124] The test power system used in this case was inspired by the Nordic32 test system and was implemented in a software tool, named Power System Simulator for Engineering (PSS/E). The power system consists of 46 nodes of which 20 are voltage controlled. Modifications were made to branch elements as to neglect resistive losses and generating units in order to represent them with identical dynamic characteristics.

[0125] A contingency analysis or assessment is conducted in PSS/E using the prior art method of time domain simulations. Typically, time domain simulations are too time consuming to perform in real-time, however, they are known to provided very precise results and therefore suitable as reference for further test methods.The cases studied reflect the total set of 33 individual N-1 cases related to loss of a single 400 kV line. Time response to every contingency was studied to determine an instant of steady-state at which a snapshot of nodal voltages could be taken. This snapshot would be used as a time domain reference for comparing with the Newton Raphson power flow method and the Thvenin Equivalent based Static Contingency Assessment, respectively.

[0126] A prior art Newton Raphson power flow method (NR) was conducted in PSS/E using the same modifications to the test power system as described above. The input scenario was identical to that used for time domain simulations except the selection of a slack-bus at which active power mismatches are balance as required in NR. NR converged in all 33 scenarios (i.e. the network converges towards a steady state in all 33 scenarios). The Newton Raphson power flow method typically evaluates the power provided to the system and whether the system is in a steady state.

[0127] A Thvenin Equivalent based Static Contingency Assessment (TESCA) according to the present disclosure is implemented in Matlab. Simulations are conducted on an input scenario composed of an admittance matrix and an initial set of nodal voltages and power injections. The input scenario is consistent with that used for time domain simulations to a precision of 10.sup.5. The Thvenin impedances and the grid transformation matrix were modified according to the 33 contingencies, and post-contingency snapshots of steady state nodal voltages were obtained. It is an advantage of the Thvenin Equivalent based Static Contingency Assessment that also the rotor angle is included in the analysis, as compared with for example the Newton Raphson power flow method.

[0128] In order to compare the nodal voltages, determined by the Newton Raphson power flow method and the method using Thvenin Equivalent based Static Contingency Assessment, respectively, a common angular reference is required. A reference node is chosen as a solid reference between the datasets originating from the Newton Raphson power flow method and the method using Thvenin Equivalent based Static Contingency Assessment, respectively, and the time domain reference. All snapshots of post contingency nodal voltages are rotated so the voltage angle at the reference node is exactly identical in all data sets. Errors between results obtained by Newton Raphson power flow method and the method using Thvenin Equivalent based Static Contingency Assessment, respectively, i.e. the methods under test, and the time domain reference cases are stated in terms of a total vector error (TVE):

[00002]$\mathrm{TVE}=\frac{.Math.{\tilde{V}}_{\mathrm{MUT}}-{\stackrel{.Math.}{V}}_{\mathrm{TD}}.Math.}{.Math.{\stackrel{.Math.}{V}}_{\mathrm{TD}}.Math.}.Math.100.Math.\%$

[0129] , where {tilde over (V)}.sub.VMUT refers to a voltage node determined by one of the methods under test and {tilde over (V)}.sub.TD refers to a voltage node determined by the result in the time domain. TVE is determined for every single voltage phasor of a snapshot.

[0130] Choice of reference node impacts the distribution of TVEs over a snapshot as any error originating from the angle of the reference phasor will be transferred to the remaining TVEs of the test system. Therefore results of Newton Raphson power flow method and the method using Thvenin Equivalent based Static Contingency Assessment, respectively, are evaluated on basis of the single largest TVE in every post-contingency snapshot.

[0131] FIG. 10 shows contingency cases ordered according to descending error of Newton Raphson power flow method results together with the corresponding maximum error of the Thvenin Equivalent based Static Contingency Assessment (TESCA). The figure shows that the Thvenin Equivalent based Static Contingency Assessment (TESCA) reproduces the time domain results with significantly better precision than Newton Raphson power flow method (NR). Of the 33 cases studied all results obtained by the Thvenin Equivalent based Static Contingency Assessment (TESCA) are within 3.0% TVE and most are within 1.0% TVE.

[0132] Therefore, an advantage of the method according to the present invention is that calculations may be reproduced with high precision, such as within 1.0% to 3.0% TVE.

[0133] FIG. 11 shows simulation results of a further embodiment of the method according to the present invention. In the further embodiment, the determining of respective Thvenin equivalents for respective voltage controlled nodes includes sequentially factorization of an admittance matrix on all non-controlled nodes and parallelization of determining Thvenin equivalents for voltage controlled nodes in a number of processors.

[0134] By parallelizing the determining of Thvenin equivalents in a number of processors improves the speed at which the static security assessment of the power system may be performed or the speed at which the contingency assessment of the power system may be performed, however at the expense of an increased load of internal communication between the processors.

[0135] The test system used in this particular example includes 2602 branches and 1648 nodes of which 313 are with voltage control. The resulting grid transformation matrix is a 313 by 313 matrix with 56478 non-zero entries.

[0136] In FIG. 11, The Algorithm 1 curve represents the further embodiment taking into account the load of internal communication between the processors, and the Amdahl curve represents the further embodiment without taking into account the load of internal communication between the processors.

[0137] For the Amdahl curve, it is seen that the speed increases almost linearly with increased number of processors, until the load of the sequentially factorization of the admittance matrix starts to dominate.

[0138] For the Algorithm 1 curve, it is seen that the speed increases up to 12 processors at which point the increase of the internal communication between the processors starts to dominate the advantage of adding more processors.

[0139] FIGS. 12A and 12B show the Nordic 32 test system and simulation results of a method according to the present invention, respectively, and where the result shows the development in voltage angle following tripping of a line between two busses.

[0140] In this specific example, the method according to the present invention is denoted as Thvenin Equivalent based Static Contingency Assessment (TESCA).

[0141] In this case, the Thvenin Equivalent based Static Contingency Assessment is applied to a test system, see FIG. 12A, for screening of Aperiodic Small-Signal Rotor Angle Stability (ASSRAS), and the applied contingency is limited to loss-of-line contingencies. The test power system used is a modification of the Nordic 32 Cigre test power system. The test power system is modified to make it prone to Aperiodic Small-Signal Rotor Angle instability by removing a generating unit from a first node, denoted as node 1021, and changing the exciter of a 200 MW unit at a second node, denoted as node 1022, to manually excite M.sub.E. In Thvenin Equivalent based Static Contingency Assessment, a manually excited machine was modelled as an internal voltage {hacek over (E)}.sub.j of constant magnitude behind a synchronous reactance X.sub.s.

[0142] Thvenin Equivalent based Static Contingency Assessment was used to identify contingencies causing aperiodic small signal instability in a case where the cause of Aperiodic Small-Signal Rotor Angle instability was due to loss of either of the lines connecting nodes 1021 and 1022. To verify the results of Thvenin Equivalent based Static Contingency Assessment, the time response of this event was simulated using PSS/E.

[0143] FIG. 12 shows the result for the voltage angles and the rotor angle for a machine, denoted as unit 1021:1. As seen in FIG. 12, at time equals to 10 seconds, one of the lines, connecting the generator at node 1021 with the remaining system, is tripped causing the rotor angle to increase. The voltage angle of the machine starts to oscillate when the rotor angle of the machine has increased to a certain level. In this specific example, the machine starts to be unstable when the rotor angle is approximately 100 (i.e. at time equals to 11.3 seconds).

[0144] Furthermore, FIG. 12 shows that the method is able to predict, by introducing a contingency into a power system that if the contingency is going to happen in real life an instable power system would be the result.

[0145] Expressions such as comprise, include, incorporate, contain, is and have are to be construed in a non-exclusive manner when interpreting the description and its associated claims, namely construed to allow for other items or components which are not explicitly defined also to be present. Reference to the singular is also to be construed as being a reference to the plural and vice versa.

[0146] A person skilled in the art will readily appreciate that various parameters disclosed in the description may be modified and that various embodiments disclosed and/or claimed may be combined without departing from the scope of the invention.