RECONSTRUCTION OF SATURATED CURRENT TRANSFORMER SIGNALS

Abstract: The invention relates to a numerical method which via discrete-time measurements of the secondary current (is(k)) from a current transformer reconstructs the primary current (ipr(k)) and the secondary current (isr(k)) of a saturated current transformer, which takes place in a block B1 and a method for detecting saturation as well as for detecting when saturation ceases, which takes place in blocks B2, B3, B4, B5, B6 and B7. The output signals of the method are obtained via a block B8 and upon a decision about saturated current transformer, the output signals consist of the reconstructed primary current (ipn(k)) and the reconstructed secondary current (isn(K)) and upon a decision about unsaturated current transformer, the output signals consist of the reconstructed primary current (ipn(k)) and the measured secondary current (is(k)). The invention also relates to a device for carrying out said methods.

Description: Reconstruction of saturated current transformer signals TECHNICAL FIELD Within largely all distribution of electric power, monitoring systems for detecting short circuits and other abnormal states are needed. Instrument transformers for measuring current and voltage therefore constitute ' an important and integrated part of control, monitoring and protective devices in most power distribution systems. It is, of course, very important that these measuring devices correctly reproduce the quantities they are designed to measure, both with regard to static and dynamic values. Since both control and monitoring devices and particularly protective devices are nowadays based on instantaneous value measurement, very high demands are placed on good dynamic conformity during the measurements. Due to practical and economic reasons it is not possible to dimension current transformers so as to avoid saturation at all fault current levels. Therefore, during a fault condition a current of such a high value may arise that the current transformer becomes magnetically saturated, whereby the waveform of the delivered current becomes distorted. This leads to considerable problems, especially for current measuring numeral protective relays, since these will then receive false information on which to base the decisions to take action. A typical example is a differential protection device which, when a current transformer saturation occurs, may lead to an external fault being incorrectly detected as an intersyste fault. It is therefore of great importance to be able to detect whether a current transformer has become subjected to such a current that it has become magnetically saturated. The present invention describes a method for reconstructing, during saturation, with the aid of the distorted secondary current, the primary current which causes the saturation. A reconstruction is also performed of the secondary current which would have been obtained unless the current transformer had become saturated. The invention also comprises a method for detecting saturation and detecting when the saturation ceases. The invention also comprises a device for carrying out the method. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 shows an equivalent diagram for a current transformer seen from the secondary side. Figure 2 shows a flow diagram according to the invention. BACKGROUND ART, THE PROBLEMS One way of reducing the effect of a current transformer saturation is to use some method for detecting saturation. Several such methods exist, which is clear, inter alia, from SE 90100917-5. When a saturation has been determined, the conditions for breaking the current are made more conservative in existing protective relays. This assumption, however, has some apparent shortcomings; for example, it means that a short-circuit current may be allowed to act for an unacceptably long period of time, which in turn may lead to negative effects on the network. A further problem in this connection is to detect when the saturation ceases. A general description of current transformers, both in steady and transient states, is given by A Wright in "Current Transformers, Their Transient and Steady State Performance", Chapman and Hall Ltd., 1968. A model which is suitable for further study of the behaviour of current transformers at different magnetizing states is described, inter alia, by T Conrad, J Schlabbach and R Speh in "Verfahren zur Korrektur der verzerrten Sekundarstrόme von Stromwandlern", etzArchiv Bd. , Vol. 6, 1984, in which it is referred to an equivalent diagram according to Figure 1 where ip = stepped down primary current i s = secondary current Lμ = magnetizing inductance Rfe = loss resistance in the iron core R2 = resistance of burden, winding, and conductors L2 = inductance of burden and leakage inductance 2 1 = current transformer ratio The differential equations for the circuit according to Figure 1 will then be It should be pointed out here that the values of both Lμ and R f e vary in dependence on the primary current via the magnetizing current. In the following description, however, Rf e will be assumed to be constant. When the magnetizing current exceeds a certain value, Lμ will decrease drastically. The greater part of the stepped down current will then be conducted via the magnetizing inductance, which results in the severely distorted secondary current. In the following description, the term "primary current" will be used only for "the stepped down primary current". It should be pointed out here that the magnetizing current iμ is needed in explicit form to be able to calculate Lμ. The differential equations (1) and (2) can be solved with numerical methods, which, inter alia, is clear from an article by T Conrad, J. Schlabbach and R Speh: "Verfahren zur Korrektur der verzerrten Sekundarstrόme von Stromwandlern" published in etzArchiv Bd. , Vol 6, pp 77-79, 1984. In these calculations a special algorithm has been used for Lμ which provides a saturated value which is only 60 times lower than the unsaturated value. The result of the calculations and the simulations which have been made show a fairly good reconstruction of the primary current. The value of the method described must, however, be considered to be relatively limited. This is due, among other thing, to the saturated value of Lμ in reality being rather 1000 times lower than the unsaturated value. Considering, in addition, the transfer functions corresponding to the differential equations, it will be seen that a high gain is required to reconstruct the primary current. This means that the method used is probably sensitive to measurement disturbances and since no disturbances are used during the simulations, the usefulness of this method in practice must be considered doubtful. Differential equations of the types according to (1) and (2) can be described in state-space form according to known technique, see, for example, T Kailath, Linear Systems, Prentice-Hall. Inc., Englewood Cliffs, N. J. 1980, pp. 50- 60. A general state-space model for the current transformer is given by dx/dt = AcCLμ.x + B c (Lμ.)i p (3) i s = Cs x (4) iμ = Cμ x ( 5 ) where it has been indicated that the state matrix A c and the vector B c depend on Lμ, .which in turn is dependent on the magnetizing current iμ. The selections of state-space representations which are utilized in the invention will be described under the summary of the invention. Equations (3) , (4) and (5) describe the model in continuous time. In "Computer Controlled Systems", p. 37, Prentice Hall, Inc., 1984, by K J Astrόm and B Wittenmark, there is described a method which discretizes the equations, whereby the following sampled state model is obtained. x(k + 1) = Ad(Lμ) x.k) + Bd( μ) i p (k) (6) i s (k) = C s x.k) (7) iμO = μ x.k) (8) Here Ad(Lμ) = eA c ( L μ ) h (9) h Bd(Lμ) = I eA c ( L μ ) s ds B c ( μ) (10) 0 where h is the sampling interval. To calculate (9) and (10), it is assumed that Lμ, which is a function of the magnetizing current iμ, is constant during the sampling interval. A model of a primary circuit upon a short circuit, which is often described in the literature (see, e.g. "Digital protective relaying through recursive least-squares identification" by A Isaksson, published in IEE Proc. Pt. C, Vol. 135, No. 5, 1988, p. 441) describes the primary current as m ip(t) = doe ~t τ + (∑. c n sin(ω c nt + Φ n ) ) (11) n=l where τ is the time constant of the circuit, m is the number of harmonic components, ω c is the fundamental frequency, and Φ n the phase shift. To describe the primary current during a short circuit, in, inter alia, "Estimation of the primary current in a saturated transformer", by K W Chen and S T Glad: Report LiTh-ISY-I 120, 1991, Department of Electrical Engineering, Linkόping University, Linkόping, Sweden, the assumption is made that the exponential term is replaced by a Taylor series expansion. However, it is obvious to use a fixed, predetermined value of τ. This reduces the number of parameters to be estimated while at the same time the model becomes less sensitive to unmodelled harmonics and noise. For a general continuous time signal i(t), the corresponding sampled value is denoted i(k), that is, i(k) =i(t) l t =kh (12) where k is a running index. It is part of the prior art that a sampled version of (11) via linear regression can be written in state-space form as ip(k) = φ T θ(k) (13) θ(k + 1) = F θ.k) (14) According to conventional recursive technique, T means transposition, θ is a parameter vector, and φ a regression vector. According to M S Sachdev, H C Wood and N C Johnson: "Kalman filtering applied to power system measurements for relaying", IEEE Trans, on PAS, Vol. PAS-104, No. 12, 1985 pp. 3565-3571, φ and F for the case m = 1 can be written as φ T _ = (1 1 0) :i5) ,-h/τ F = 0 cos(ω c h) sin(G_ c h) (161 -sin(G)ch) cos (ω c h) Now, if the output signal i p (k) from the signal model according to (13) and (14) is taken as input signal to the current transformer model according to (6), (7) and (8), the resultant system according to known technique (see, e.g., "Control System Toolbox", The Math Works, Inc, p. 2.149) can be written as z(k + 1) = A z(k) (17) ip(k) = Cp z(k) (18) i s (k) = C s t z(k) (19! iμ(k) = Cμt z (k) 120) where Cp t = (φ τ 0 o; 122! C s t = (0 0 0 C s ) (23 Cμ t = (0 0 0 Cμ) (24 ) and where the used state vector is given by z(k) = (θ(k)T x(k) τ ) τ (25) In similar model contexts it is common to use a Kalman filter to reconstruct the state vector z(k) . The general method for such filtering is described, inter alia, in "Optimal Filtering", by B D O Anderson and J B Moore, Prentice Hall, 1979, p. 44. If a fixed value of Lμ is used, A in equation (17) will become a constant matrix. It is then possible to utilize a stationary Kalman filter, which provides the reconstruction z(k) = t __(k-l) + K( i s (k) Cst A t z(k-l) ) (26) Here, "K" is a gain vector. One way of calculating "K" is shown in B Carlsson, "Digital Differentiating Filters and Model Based Fault Detection" , PhD thesis , Department of Technology, Uppsala University, Uppsala, Sweden, 1989 , pp . 72-76. In a report LiTh-ISY-I 120 from the Department of Electrical Engineering, Linkόping University, Linkόping entitled "Estimation of the primary current in a saturated transformer" , 1991, presented by K W Chen and S T Glad, a method is described based on a Kalman filter applied to a signal model which is based on Taylor series expansion and connected to a current transformer model . Otherwise , continuous time is used here with a time-varying Kalman • gain, which per se requires a high numerical capacity. To detect saturation, a simple detector is used, based on the variance of the estimated fault . SUMMARY OF THE INVENTION The invention relates to a numerical method which, via discrete-time measurements of the secondary current i s (k) from a current transformer, reconstructs the primary current and the secondary current in case of saturation of current transformers and a method for detecting saturation as well as for detecting ' when saturation ceases. The reconstruction takes place with the aid of current transformer models of an unsaturated and a saturated current transformer as well as a signal model of the primary current, wherein the the current transformer models are fed from the signal model and saturation is decided by a decision logic unit when the output signal from the saturated current transformer model gives a better description of the secondary current than what the output signal from the unsaturated current transformer model gives. The output signals i pr (k) and i sr (k) in case of a decision about saturated current transformer consist of a reconstructed primary current i pn (k) and a reconstructed secondary current i sn (k) and, in case of a decision about unsaturated current transformer, of a reconstructed primary current ip n d and an actually measured secondary current i s (k) . The invention also relates to a device for carrying out the above-mentioned methods. DESCRIPTION OF THE PREFERRED EMBODIMENTS The invention is based on the current transformer model which is described by equations (1) and (2) and the general state-space model described by equations (3), (4) and (5). The following state-space description is used in equations (3) , (4) and (5) A c (Lμ) = (27) C s = (0 1) (29: Cμ = (1 0) (30) Further, the invention is based on discretization according to equations (9) and (10), a sampled signal model according to equations (13) ... (16), and on the above-described combined signal and current transformer model which is described by equations (17) ... (25) . According to the invention, two values of the inductance Lμ are used, which is denoted kμ = the nominal value of the magnetizing inductance μs = the saturated value of the magnetizing inductance This means that according to the invention two state-space descriptions or models are obtained, whereby also the matrix t according to equation (21) is given two values according to 0 Ant = Bd ( Lμn ) φ T Ad ( μn ) (31) N ow, operating with two different values of the inductance it means at the same time that two different values of A - \ obtained according to the follow i ng and B d are (9n) Ad(Lμn ) = eA c ( L μn )h (lOn) Bd ( Lμn ) = i e A c ( Lμ n . sds B c ( Lμn ) (10s) j e A c ( L μs )S ds B c ( Lμs ) B ( Lμs ) ) = 5 B ( :( μs) = ( The numerical method according to the invention completes a number of steps for each sampled measurement of the secondary curren . The different steps comprise reconstruction from an unsaturated model of a current transformer, reconstruction from a saturated model, detection of saturation, as well as decision about returning from a saturated to an unsaturated state. When saturation is not detected, a parallel prediction takes place with both the unsaturated and the saturated current transformer model. With the unsaturated model, the reconstruction is performed from a Kalman filter with a pre-calculated gain K, with A nt according to equation (31) , and with the C-parameters according to equations (22) and (23) as follows: _z n (k) = A nt ' z n (k-l) + -g-(i s (k) - C st A nt __ n (k-l)) (33) i pn (k) = C pt Zn( ) (34) i sπ (k) = C st £n(k) (35) Here, i pn (k) is a reconstruction of the primary current and i s n(k) is a reconstruction of the secondary current. The invention is further based on the fact that the effect of i s (k) on the state reconstruction z n (k) is shut off as soon as a saturation has occurred. This is done with the aid of the gain parameter "g" which, the first time a saturated state is detected, is changed from 1 to 0. This means that during saturation, the currents will be reconstructed based on measurements of the secondary current prior to saturation. To detect that saturation has occurred, a saturated model according to the invention can be used and the following relationships be utilized: _ s (k) = (A st ) d+1 £n(k - d -1) (36) i ss (k) = Cst ^s(k) (37) with A st according to equation (32) . The variable "d" will be defined below after equation (40) . As is clear, the invention calculates and uses a state reconstruction z s (k) for the saturated model which is determined by the state reconstruction ^ n (k-d-1) , that is, a state reconstruction obtained d+1 samples earlier. According to the invention, the reconstruction error for the respective model is then calculated according to e n = is(k) - i sn (k) (38) e s = i s (k) - iss(k) (39) In the following the invention will now be described with reference to a flow diagram according to Figure 2. The reconstruction method as described above and in accordance with equations (27) ... (39) takes place in a block Bl, which is supplied with the sampled discrete-time secondary current i s (k) . Next, a comparison is made between the absolute value of the reconstruction errors for the two models. According to the flow diagram in Figure 2, this takes place in block B2. If the reconstruction error calculated on the basis of the saturated model is smaller than the reconstruction error for the unsaturated model, that is, if I e s I < I e n i (40) this indicates that a saturation has occurred. To prevent short disturbances from denoting a saturated state, it is required according to the invention that the inequality remains for a predetermined number (f) of consecutive samples for saturation to be determined. The control whether the saturation indication has existed during the above- mentioned number of samples is made by storing, in a variable "d" in the block B3, the number of consecutive times that equation (40) is true. This variable is set to zero if equation (40) is false. The first time equation (40) is true, that is, when d=l, the gain parameter g will be changed from g = 1 to g = 0, which is made in the block B4, which then restores this information to the block Bl, equation (33), to prevent the distorted current signal from being allowed to influence the reconstruction. Now, if the counter for the variable d shows that equation (40) is true for a number of consecutive samples d > f according to block B5, a decision about saturation is made, which takes place in the block B6 where also the variable M is set equal to 0. If the saturation indication determines that no saturation has occurred, block B7 will generate M = 1, which is passed to a block B8. To this block are supplied the sampled secondary current i s (k) , the primary current i pn (k) reconstructed by block Bl as well as the reconstructed secondary current i sn (k) . By an M = 1 from block B7, block B8 will deliver a value of the primary current i pr (k) which is equal to the reconstructed primary current i pn (k) and a value of the reconstructed secondary current i sr (k) which is equal to the measured secondary current i s (k). Now, if the saturation indication remains for the preset number of samples f, the method according to the invention will understand that a saturation, -that is, M = 0 according to the flow -diagram, has occurred. The output signal i pr (k) from block B8 will then be represented by the reconstructed primary current i pπ (k) and the output signal i sr (k) by the reconstructed secondary current i sn (k) . The iteration goes on continuously in the usual way by setting k = k + 1, shown in principle by block B9. As will be clear from the above, according to the invention a preferred embodiment with two current transformer models is used, one representing an unsaturated model and the other a saturated model. Within the scope of the invention, several current transformer models, based on fixed magnetizing inductances with a value which lies between those corresponding to a nominal and a saturated value, may be used. Within the scope of the invention, the signal model may also consist of harmonics according to equation (11) . Within the scope of the invention, a variant of an embodiment with regard to detection of saturation, as described by means of blocks B2, B3 and B4, may be replaced by another detection principle, for example a method described in SE 9100917-5. It is, of course, also important to be able to detect when a saturation ceases. One embodiment is described by the inequality (40) and the block B2, respectively. Also in this case, similar detection principles, capable of replacing the contents of blocks B2, B3 and B5, fall within the scope of the invention. As an example, the reconstructed magnetizing current iμ, when saturation has been determined, may be stored. During saturation, the magnetizing current will be practically equal to the reconstructed primary current i p (k). This means that the magnetizing current increases drastically during saturation. Information to the effect that saturation has ceased may be obtained when the magnetizing current decreases below the stored value. It is, of course, also possible to use i s (k) directly to make a decision about restoration to an unsaturated state. A device for carrying out the method according to the invention preferably forms an integral part of a protective relay where current measurement takes place with the aid of current transformers- From the design point of view, the device may be arranged in the form of a number of blocks according to Figure 2 or constitute an integrated device. In an integrated embodiment all equations and logical decisions are included in a computer-based program which has the discrete-time, sampled secondary current i s (k) as input signal and whose output signals, upon a decision on an unsaturated state, consist of ipr(k) = ipn(k) and isr(k) = i s (k) and whose output signals, upon a decision on a saturated state, consist of i pr (k) = ipn(k) and i S r(k) = isn(k).

Claims: CLAIMS 1. A method for reconstruction of saturated current transformer signals, characterized in that the reconstructed current transformer signals which comprise reconstructed primary current (i pr (k)) and reconstructed secondary current (i sr (k)) are obtained via sampled discrete-time measurement of the secondary current (i s (k)) and with the aid of at least two current transformer models and a signal model of the primary current. 2. A method for reconstruction of saturated current transformer signals according to claim 1, characterized in that the reconstruction is obtained with the aid of current transformer models of a saturated and an unsaturated current transformer. 3. A method for reconstruction of saturated currrent transformer signals according to claim 1, characterized in that the current transformer models are fed from the signal model. 4. A method for reconstruction of saturated current transformer models according to claim 1, characterized in that saturation is decided by a decision logic unit when the output signal from the saturated current transformer model gives a better description of the secondary current than what the output signal from the unsaturated current transformer model gives. 5. A method for reconstruction of saturated current transformer signals according to claims 1 to 4 and wherein the reconstruction and the saturation decisions are based on a combination of the signal model and the unsaturated and the saturated current transformer model and wherein the method is characterized by the following relationships ' z n (k) = Ant zn(k-l) + K-g- (i s (k) - C s t A nt " zn(k-1)) (33) ipn(k) = C p t z n (k) (34) i S n (k) = Cs z n (k) (35) where z n (k) is the state reconstruction based on an unsaturated current transformer model Ant = (31) Bd ( Lμn ) φ T Ad ( Lμn ) ,-h/τ F = cos(__ c h) sin(ω c h) (16) -sin(ω c h) cos(ω c h) Bd(Lμn) = J eA c ( Lμ n ) s ds B c ( μn) (lOn) 0 ω c is the frequency of the fundamental tone Lμ is the nominal magnetizing inductance φ J = (1 1 0) (15) Ad ( Lμn ) = e A c (L μn )h (9n) Bc(L ^ )= ( & i? (28n) Rfe = loss resistance in the iron core R2 = resistance of burden, winding, and conductors L2 = inductance of burden and leakage inductance K is a predetermined gain to the reconstruction according to equation (33) g is a gain parameter which may assume the value 0 or 1 and which, upon a decision about unsaturated state, assumes the value 1 ipn(k) is a reconstructed primary current i sn (k) is a reconstructed secondary current Cpt = (φ T 0 0) ( 22 ) C st = (0 0 0 C s ) (23) C s = (0 1) (29) h = sampling interval τ = time constant of primary circuit and for detecting saturation, the following reconstruction, based on a saturated current transformer model, is used zs(k_ = ( Ast ) d+1 ^n(k - d -1) (36) i ss (k) = Cst £ s (k) (37; where ' zs (k) is the state reconstruction based on a saturated current transformer model Lμ s is the magnetizing inductance for a saturated current transformer d( μs) = e A c (L μs )h (9s) Bd (Lμ s ) = J e A c ( Lμ s ) s ds B c (L μs ) (10s) 0 (28s) after which the reconstruction error for the two models is determined according to e n = is(k) - isn(k) (38) e s = i s (k) - i SS (k) (39) and when 10 I e s I < I en I (40) is true, a variable "d" is adapted to store the number of consecutive true samples 15 and when d = 1, g is changed from 1 to 0 and if d > f, where "f" is a predetermined consecutive number of samples, it is decided that saturation has occurred, whereby the measured secondary current i s (k) is 20 replaced by the reconstructed secondary current i sn (k) and when equation (40) is no longer true, or if the predetermined number of consecutive samples "f" is not achieved, the current transformer is considered not to be 25 saturated. 6. A device for carrying out the method for reconstruction of saturated current transformer signals according to claims 1-5, characterized in that the device comprises a block 30 Bl which has the sampled discrete-time secondary current i s (k) as input signal and which is programmed with the equations Ad(Lμ) = eA c (L μ )h (9) 35 Bd(Lμ. = J e A c (L μ ) s ds B c (Lμ) (10) 0 A c (Lμ) = (27) B c (Lμ) = R R T (28) (t,x) Z.2 C s = (0 1) (29) Cμ = (1 0) (30] Ant = (31) Bd ( Lμn ) φ T Ad (Lμn) A s = (32) BdCL μs )φT Ad(Lμ s ), z n Ck ) = A nt z n (k-1) + K-g- (i s (k) - C s A n t z n (k-l )) (33) ipn(k) = C pt z n (k) (34) i sn (k) = C s t z n (k) (35) z s (k) = (A st ) d+1 zn(k - d -1) (36) i ss (k) = C st z s (k) (37) e n = i≤(k) - i S n(k) (38) e s = i s (k) - i S s(k) (39) and with corresponding equations (9n) , (9s) , (lOn) , (10s) , (27n) , (27s), (28n) and with variables included defined according to claim 5 and that the values of e n and e s obtained in block Bl are arranged as input values to a block B2 which is adapted to determine whether I e s I < I e n I (40) is true, and if this is the case, information about this is adapted to be supplied to a block B3 which stores the number of consecutive samples that equation (40) is true in a variable "d" and the first time that d = 1, this information is adapted to be supplied to a block B4 which is adapted to change the variable "g" in equation (33) from g = 1 to g = 0, which change is adapted to be returned to block Bl and that the consecutive value stored in the variable "d" in block B3 is adapted to be supplied to a block B5 adapted to check whether d > f where "f" is a preset value, and if this is the case, information about this is adapted to be supplied to a block B6 for decision that saturation has occurred, whereby a variable M is set equal to 0 and that if equation (40) is false, information about this together with information as to whether the inequality of block B5 is false are adapted to be supplied to a block B7 for information that no saturation has occurred, whereby the variable M is set equal to 1 and that the current value of the variable M is adapted to be supplied to a block B8 together with the sampled measured value i s (k) of the secondary current, the primary current ipn(k) reconstructed by block Bl and the reconstructed secondary current i sn (k) and that when M = 0, block B8 is adapted to deliver a value ipr(k) of the primary current which is equal to .the reconstructed value i pn (k) of the primary current and a value i S r(k) of the secondary current which is equal to the reconstructed value i sn (k) of the secondary current and that when M = 1, block B8 is adapted to deliver a value ipr(k) of the primary current which is equal to the reconstructed value i pn (k) of the primary current and a value i sr (k) of the secondary current which is equal to the sampled measured value i s (k) and that the iteration is adapted to continue by setting, in a block B9, k = k + 1.

Inventors: BJOERKLUND MAGNUS, CARLSSON BENGT, MOHAN SAHA MURARI, BJOERKLUND, MAGNUS, CARLSSON, BENGT, MOHAN SAHA, MURARI
Applicants: ASEA BROWN BOVERI, ASEA BROWN BOVERI AB
Application year: 1992
Main classification: H01F40
IPC codes: H01F27, H02H1