Determination of power transmission line parameters using asynchronous measurements

Abstract

There is described a method of determining power transmission line parameters using non-synchronous measurements acquired from different locations along a power transmission line. The method comprises (a) acquiring first measurement data comprising corresponding pairs of voltage and current measurement values measured at a first location along the power transmission line at predetermined points in time relative to a first time reference, (b) acquiring second measurement data comprising corresponding pairs of voltage and current measurement values measured at a second location along the power transmission line at predetermined points in time relative to a second time reference, (c) calculating initial values of the power transmission line parameters and an initial value of the difference between the first time reference and the second time reference based on the first measurement data and the second measurement data, and (d) calculating resulting values of the power transmission line parameters and a resulting value of the difference between the first time reference and the second time reference by utilizing a least squares algorithm and the initial values of the power transmission line parameters, the initial value of the difference between the first time reference and the second time reference, the first measurement data, and the second measurement data. Furthermore, a data acquisition device and a system for determining power transmission line parameters as well as a computer program and a data carrier are described.

Claims

1. A method of determining power transmission line parameters using non-synchronous measurements acquired from different locations along a power transmission line, the method comprising acquiring first measurement data comprising corresponding pairs of voltage and current measurement values measured at a first location along the power transmission line at predetermined points in time relative to a first time reference, acquiring second measurement data comprising corresponding pairs of voltage and current measurement values measured at a second location along the power transmission line at predetermined points in time relative to a second time reference, calculating initial values of the power transmission line parameters and an initial value of the difference between the first time reference and the second time reference based on the first measurement data and the second measurement data, and calculating resulting values of the power transmission line parameters and a resulting value of the difference between the first time reference and the second time reference by utilizing a least squares algorithm and the initial values of the power transmission line parameters, the initial value of the difference between the first time reference and the second time reference, the first measurement data, and the second measurement data, wherein the power transmission line parameters comprise series conductance, series susceptance, and shunt capacitance.

2. The method according to claim 1, wherein the initial value of the series conductance (G) and the initial value of the series susceptance (B) are calculated by calculating a first admittance based on the first measurement data, calculating a second admittance based on the second measurement data, calculating a series impedance (Z.sub.S) based on the first admittance and the second admittance, calculating the initial value of the series conductance as the real part of the reciprocal of the series impedance, and calculating the initial value of the series susceptance as the imaginary part of the reciprocal of the series impedance.

3. The method according to claim 2, wherein the initial value of the shunt capacitance is calculated by applying sensitivity analysis with regard to active power loss and reactive power loss.

4. The method according to claim 2, wherein the initial value of the difference between the first time reference and the second time reference is calculated as a difference between the phase of the voltage of the second measurement data and the phase of a voltage calculated on the basis of the first measurement data and the initial value of the series conductance, the initial value of the series susceptance, and the initial value of the shunt capacitance.

5. The method according to claim 1, wherein the initial value of the shunt capacitance is calculated by applying sensitivity analysis with regard to active power loss and reactive power loss.

6. The method according to claim 1, wherein the initial value of the difference between the first time reference and the second time reference is calculated as a difference between the phase of the voltage of the second measurement data and the phase of a voltage calculated on the basis of the first measurement data and the initial value of the series conductance, the initial value of the series susceptance, and the initial value of the shunt capacitance.

7. The method according to claim 1, wherein the first measurement data is acquired by a first data acquisition device positioned at the first location, the second measurement data is acquired by a second data acquisition device positioned at the second location, and the steps of calculating initial values and resulting values is carried out by the first data acquisition device after receiving the second measurement data from the second data acquisition device.

8. The method according to claim 7, further comprising sending a start signal from the first data acquisition device to the second data acquisition device, sending a response signal from the second data acquisition device to the first data acquisition device, determining a communication roundtrip time, and continuing with the calculation of initial values and resulting values if the communication roundtrip time is below a predetermined threshold value.

9. A computer program comprising non-transitory computer executable instructions, which, when executed by a processor of a computer, are configured to carry out the method according to claim 1.

10. A data carrier loaded with the computer program according to claim 9.

11. A data acquisition device for determining power transmission line parameters, the data acquisition device comprising a clock unit adapted to provide a time reference, a data acquisition unit adapted to acquire measurement data comprising corresponding pairs of voltage and current measurement values at a location along a power transmission line at predetermined points in time relative to the time reference, a data communication unit adapted to receive remote measurement data from a remote data acquisition device, the remote measurement data comprising corresponding pairs of voltage and current measurement values measured at a remote location along the power transmission line at predetermined points in time relative to a remote time reference, and a data processing unit adapted to calculate initial values of the power transmission line parameters and an initial value of a difference between the time reference and the remote time reference based on the measurement data and the remote measurement data, and calculate resulting values of the power transmission line parameters and a resulting value of the difference between the time reference and the remote time reference by utilizing a least squares algorithm and the initial values of the power transmission line parameters, the initial value of the difference between the time reference and the remote time reference, the measurement data, and the remote measurement data, and wherein the power transmission line parameters comprise series conductance, series susceptance, and shunt capacitance.

12. A system for determining power transmission line parameters using non-synchronous measurements acquired from different locations along a power transmission line, the system comprising a first data acquisition device adapted to acquire first measurement data comprising corresponding pairs of voltage and current measurement values measured at a first location along the power transmission line at predetermined points in time relative to a first time reference, a second data acquisition device adapted to acquire second measurement data comprising corresponding pairs of voltage and current measurement values measured at a second location along the power transmission line at predetermined points in time relative to a second time reference, and a data processing device adapted to calculate initial values of the power transmission line parameters and an initial value of the difference between the first time reference and the second time reference based on the first measurement data and the second measurement data, and calculate resulting values of the power transmission line parameters and a resulting value of the difference between the first time reference and the second time reference by utilizing a least squares algorithm and the initial values of the power transmission line parameters, the initial value of the difference between the first time reference and the second time reference, the first measurement data, and the second measurement data, and wherein the power transmission line parameters comprise series conductance, series susceptance, and shunt capacitance.

13. The system according to claim 12, wherein the first data acquisition device is a phasor measurement unit, the first time reference is GPS time, and the first data acquisition device is adapted to use the resulting value of the difference between the first time reference and the second time reference to synchronize the second data acquisition device to the first time reference.

Description

BRIEF DESCRIPTION OF THE DRAWING

(1) FIG. 1 shows a flow diagram of a method according to an embodiment of the present invention.

(2) FIG. 2 shows a circuit diagram of a power transmission line model as used in accordance with an embodiment of the present invention.

DETAILED DESCRIPTION

(3) The illustration in the drawing is schematic. It is noted that in different figures, similar or identical elements are provided with the same reference numerals or with reference numerals which differ only within the first digit.

(4) FIG. 1 shows a flow diagram of a method according to an embodiment of the present invention. More specifically, FIG. 1 shows the flow of messages and data between a first data acquisition device 10 and a second data acquisition device 20. The first data acquisition device 10 is arranged at a first location along a power transmission line (e.g. at one end of the transmission line) and acts as a master in this embodiment. The second data acquisition device 20 is arranged at a second location along the power transmission line (e.g. at the other end of the transmission line) and acts as a slave in this embodiment. The first data acquisition device 10 is adapted to perform a sequence of measurements to acquire first measurement data comprising vectors of voltage and current measurement values (samples). These measurements are carried out at predetermined points in time relative to a first time reference, such as an internal clock in the first data acquisition device 10. The internal clock may or may not be synchronized with GPS time. Similarly, the second data acquisition device 20 is adapted to perform a sequence of measurements to acquire vectors of voltage and current measurement values (samples). These measurements are carried out at predetermined points in time relative to a second time reference, such as an internal clock in the second data acquisition device 20. A data communication link (not shown) is present and allows the first data acquisition device 10 and the second data acquisition device 20 to communicate with each other. The data communication link may be any kind of wired or wireless link.

(5) The method begins at S0 where the first (master) data acquisition device 10 receives a request for obtaining power transmission line parameter values, in particular values of series conductance, series susceptance, and shunt capacitance. Upon receiving the request, the first data acquisition device 10 begins, at S1, to acquire the first measurement data and, at S2, sends a start signal to the second (slave) data acquisition device 20. Upon receiving the start signal, the second data acquisition device 20 begins, at S3 to acquire the second measurement data and sends, at S4, a response signal to the first data acquisition device 10. Upon receiving the response signal from the second data acquisition device 20, the first data acquisition device 10 calculates a communication round trip time at S5 and checks that the communication round trip time is below a predetermined threshold, in particular below 1/f, where f is the power network frequency. In case of a 50 Hz power network frequency, the predetermined threshold equals 20 ms. If the check confirms that the communication round trip time is below the threshold, the method continues with S61 to S6n, where a total of n sets of measurement data is transmitted from the second data acquisition device 20 to the first data acquisition device 10. Upon receiving the measurement data from the second data acquisition device 20, the first data acquisition device 10 processes the first and second measurement data at S7 to obtain the desired power transmission line parameter values. The resulting parameter values are output at S8.

(6) Now, the processing at S7 referred to above will be described in more detail with reference to the circuit diagram of a power transmission line model shown in FIG. 2. Here, the first data acquisition device 10 is located at the far left node 1 and the second data acquisition device 20 is located at the far right node 2. Accordingly, the first data acquisition device 10 measures the voltage U.sub.1 and the current I.sub.1, while the second data acquisition device 20 measures the voltage U.sub.2 and the current I.sub.2. The power transmission line is modeled as a series impedance Z.sub.S consisting of a resistance R and an inductance L, i.e. Z.sub.S=R+jωL, and a shunt capacitance C distributed in equal parts, i.e. C/2, at both ends of the transmission line.

(7) First, initial values are calculated for the sought transmission line parameters G (series conductance), B (series susceptance), and C (shunt capacitance), and for the difference θ between the first and second time references.

(8) More specifically, the initial values of G and B are calculated by ignoring the shunt capacitances C/2 (as indicated by dashed lines 11 and 22 in FIG. 2) and calculating a first admittance Y′.sub.1 as the ratio between the measured current phasor I.sub.1 and the measured voltage phasor U.sub.1 at node 1, i.e. Y′.sub.1=I.sub.1/U.sub.1. Similarly, a second admittance Y′.sub.2 is calculated as Y′.sub.2=−I.sub.2/U.sub.2. It follows from FIG. 2 that 1/Y′.sub.1=Z.sub.S+1/Y′.sub.2, such that Z.sub.S=1/Y′.sub.1−1/Y′.sub.2. The initial values of G and B are then calculated as G=real(1/Z.sub.S) and B=imag(1/Z.sub.S).

(9) The initial value of C is calculated iteratively by applying sensitivity analysis. Prior to the iterative procedure, the following variables are set: i=1, C=ΔC=10.sup.−7, B.sub.C=ωC, ΔP.sub.L.sup.(0)=R.Math.|I.sub.S|.sup.2, ΔQ.sub.L.sup.(0)=X.Math.|I.sub.S|.sup.2−(B.sub.C/2).Math.(|U.sub.1|.sup.2+|U.sub.2|.sup.2). Here, i denotes the iteration counter, ΔC is a step size, B.sub.C is the shunt susceptance, ΔP.sub.L.sup.(0) is the active power loss across the transmission line, ΔQ.sub.L.sup.(0) is the reactive power loss across the transmission line, and X is the reactance, i.e. X=ωL. Furthermore, I.sub.S is the current through the series impedance Z.sub.S (see FIG. 2) and is given as I.sub.S=I.sub.1−U.sub.1.Math.B.sub.C/2.

(10) Now the iterative procedure is performed as follows:

(11) $\mathrm{do}$$\mathrm{Find}{Z}_s^{\left(i\right)}=\frac{1}{Y_1^{\prime }-j\frac{B_c^{\left(1\right)}}{2}}-\frac{1}{Y_2^{\prime }-j\frac{B_c^{\left(1\right)}}{2}}$$\mathrm{Obtain}{\Delta P}_L^{\left(i\right)}=R.Math.{.Math.I_s.Math.}^2\mathrm{and}{\Delta Q}_L^{\left(i\right)}=X{.Math.I_s.Math.}^2-\frac{B_c^{\left(1\right)}}{2}\left({.Math.U_1.Math.}^2+{.Math.U_2.Math.}^2\right)$$\mathrm{Determine}\mathrm{coefficients}{k}_p=\frac{{\Delta P}_L^{\left(i\right)}-{\Delta P}_L^{\left(i-1\right)}}{\Delta C}\mathrm{and}$$k_q=\frac{{\Delta Q}_L^{\left(i\right)}-{\Delta Q}_L^{\left(i-1\right)}}{\Delta C}$$\mathrm{Resolve}\mathrm{capacitance}\mathrm{coefficient}k=\min \left(k_p,k_q\right)$$\mathrm{Compute}C=k.Math.\Delta C;B_c=\mathrm{\varpi C}$$i=i+1$$\mathrm{while}.Math.{\Delta P}_L^{\left(i\right)}-{\Delta P}_L^{\left(i-1\right)}.Math.>\xi \mathrm{or}.Math.{\Delta Q}_L^{\left(i\right)}-{\Delta Q}_L^{\left(i-1\right)}.Math.>\xi$

(12) The tolerance ξ is a predetermined value, such as ξ=10.sup.−6.

(13) Thus, the iterative process continues as long as the difference between successive values of the active power loss or the difference between successive values of the reactive power loss is larger than the tolerance ξ. In other words, the iterative process continues until both difference are less than or equal to the tolerance ξ.

(14) The last initial value, i.e. the time difference θ is calculated by using the measured values at node 1 and a calculated value of the voltage drop U.sub.S across the series impedance Z.sub.S to calculate an expected phasor U′.sub.2 for the voltage at node 2 and finding the phase difference between this expected voltage U′.sub.2 and the measured voltage U.sub.2, i.e. θ=arg(U.sub.2)−arg(U′.sub.2), where U′.sub.2=U.sub.1+U.sub.S=U.sub.1+I.sub.S.Math.Z.sub.S.

(15) Having obtained the above initial values of G, B, C and θ, a least squares algorithm is now performed to find more exact (or resulting) values of these parameters. More specifically, the variable sought is the vector x=[G B C θ].sup.T and the objective function f(x) is the complex currents I.sub.1 and I.sub.2 at the two nodes 1 and 2, which are given by the following equations:
I.sub.1.sup.re=G.Math.U.sub.1.sup.re−(C−B)U.sub.1.sup.im−(G cos θ−B sin θ)U.sub.2.sup.re−(G sin θ+B cos θ)U.sub.2.sup.im
I.sub.1.sup.im=G.Math.U.sub.1.sup.im+(C−B)U.sub.1.sup.re−(G cos θ−B sin θ)U.sub.2.sup.im+(G sin θ+B cos θ)U.sub.2.sup.re
I.sub.2.sup.re=G.Math.U.sub.2.sup.re−(C−B)U.sub.2.sup.im−(G cos θ−B sin θ)U.sub.1.sup.re+(G sin θ+B cos θ)U.sub.1.sup.im
I.sub.2.sup.im=G.Math.U.sub.2.sup.im−(C−B)U.sub.2.sup.re−(G cos θ−B sin θ)U.sub.1.sup.im−(G sin θ+B cos θ)U.sub.1.sup.re

(16) Before starting the iterative process, the iteration counter i is set to 1 and the initial value of the variable x is set to x.sup.(0)=[G B C θ].sup.T, where G, B, C and θ are the initial values as previously determined.

(17) Then the least squares algorithm is performed as follows:

(18) $\mathrm{do}$$\mathrm{Calculate}\mathrm{Jacobian}\mathrm{matrix}{H}_{k,j}=\frac{\partial f_k\left(x^{\left(i-1\right)}\right)}{\partial x_j};$$k=1,2,\mathrm{.Math.},m;j=1,2,3,4$$\text{//}\mathrm{where}m\mathrm{is}\mathrm{number}\mathrm{of}\mathrm{measurements}\mathrm{and}m>=4$$\mathrm{Compute}{b}_k=f_k\left(x^{\left(i-1\right)}\right)$$\mathrm{Solve}\mathrm{the}\mathrm{linear}\mathrm{problem}\Delta x={\left(H^{T}H\right)}^{-1}\left(H^{T}b\right);$$x^{\left(i\right)}=x^{\left(i-1\right)}+\Delta x$$i=i+1$$\mathrm{while}i<\mathrm{maxIter}$

(19) For a given value of k, the Jacobian matrix H is

(20) $H=\left[\begin{array}{cccc}\frac{\partial I_1^{\mathrm{re}}}{\partial G}& \frac{\partial I_1^{\mathrm{re}}}{\partial B}& \frac{\partial I_1^{\mathrm{re}}}{\partial C}& \frac{\partial I_1^{\mathrm{re}}}{\partial \theta }\\ \frac{\partial I_1^{\mathrm{im}}}{\partial G}& \frac{\partial I_1^{\mathrm{im}}}{\partial B}& \frac{\partial I_1^{\mathrm{im}}}{\partial C}& \frac{\partial I_1^{\mathrm{im}}}{\partial \theta }\\ \frac{\partial I_2^{\mathrm{re}}}{\partial G}& \frac{\partial I_2^{\mathrm{re}}}{\partial B}& \frac{\partial I_2^{\mathrm{re}}}{\partial C}& \frac{\partial I_2^{\mathrm{re}}}{\partial \theta }\\ \frac{\partial I_2^{\mathrm{im}}}{\partial G}& \frac{\partial I_2^{\mathrm{im}}}{\partial B}& \frac{\partial I_2^{\mathrm{im}}}{\partial C}& \frac{\partial I_2^{\mathrm{im}}}{\partial \theta }\end{array}\right]$
and the residual vector b is

(21) $b=\left[\begin{array}{c}{I}_{1_M}^{\mathrm{re}}-I_1^{\mathrm{re}}\\ I_{1_M}^{\mathrm{im}}-I_1^{\mathrm{im}}\\ I_{2_M}^{\mathrm{re}}-I_2^{\mathrm{re}}\\ I_{2_M}^{\mathrm{im}}-I_2^{\mathrm{im}}\end{array}\right].$

(22) In the residual vector b, the current values with index 1M and 2M are the measured values.

(23) When the iteration index i reaches the maximum value maxIter, the algorithm ends and the last value of the vector x contains the resulting values of G, B, C and θ.

(24) By applying the method described above, it is possible to provide highly accurate estimates of power transmission line parameters using asynchronous measurements, i.e. without expensive PMUs with time references fixed to GPS time.

(25) It is noted that the term “comprising” does not exclude other elements or steps and the use of the articles “a” or “an” does not exclude a plurality. Also elements described in association with different embodiments may be combined. It is further noted that reference signs in the claims are not to be construed as limiting the scope of the claims.