algorithm consists of three components:
CS sampling in PMU
CS reconstruction in control center PDC
Modified SP algorithm
In section 4.1.1, it is shown that how traditional CS sampling can be adapted for synchrophasor communication considering C37.118.1-2011 [14]. Section 4.1.2 describes
when and how often synchrophasor reconstruction can be performed in PDCs. Section 4.1.3 proposes modified Subspace Pursuit (SP) algorithm for CS reconstruction.
In Figure 4.1, block diagram of CS based synchrophasor data communication is presented. Synchrophasors are compressive sampled at a PMU and then transmitted to a super-PDC (control center PDC) directly or via substation PDCs. Synchrophasors are reconstructed at higher rates in super-PDC. The reconstructed synchrophasors are used in WAMS applications.
y
y
PDCOutput
PDC
y
y
Figure 4.1: Block diagram of CS for synchrophasor communication
4.1.1
CS Sampling in PMU
In traditional CS, signals are processed (sampling and reconstruction) in blocks. As a result, there is a waiting time delay in recovering the signal. As per standard C37.118.1- 2011 [14], it is desired to send synchrophasors as soon as they are generated. So, partial Fourier/DCT matrix has been chosen as measurement matrix in this work. The sensing matrix has one non-zero element of value 1 (spike or Dirac Delta function) in each row. The position of non-zero value is the index of the synchrophasor being reported. Due to this structure of sensing matrix, synchrophasors are sent as soon as they are generated (not in batch) over communication networks.
Transmit
Receive in Super-PDC Reconstruction in Super-PDC Compressive Sampling in
In the proposed algorithm, PMU computes synchrophasors in a random time-sequence. The random time sequence is derived from a uniformly spaced time instants. However, it is difficult to generate sequences which are ‘random’ in true sense. Computer generated random numbers or sequences are basically ‘pseudo random’ numbers or sequences. In case 1 of section 5.3, synchrophasors y1,… y15, y19, y20, y23, y25, y29, y34, y36, y39, y42,… corresponding to time instants t1,… t15, t19, t20, t23, t25, t29, t34, t36, t39, t42,… are sent to control center PDC. The time instants (t1,… t15, t19, t20, t23,…) of synchrophasors are derived from equally spaced time instants t1, t2, t3,….tt. In control center PDC, synchrophasors are reconstructed at equally spaced time instants t1, t2, t3,….tt.
Once PMU completes synchrophasor computations corresponding to a time instant, synchrophasors are sent to PDC in data packets satisfying standard C37.118.2-2011 [17]. Each synchrophasor data packet contains measurement time (time instant) of each synchrophasor along with synchrophasor value. Control center PDC knows the sampling time instant of each synchrophasor from the data packet sent by PMU. Power system synchrophasors are complex numbers. In PMU data packets, complex valued synchrophasors are represented as magnitude-angle or real-imaginary numbers [17].
4.1.2
CS Reconstruction in PDC
In the proposed algorithm, CS reconstruction is incremental in nature. In control center PDC, reconstruction process starts every time new synchrophasor (CS sample) arrives at receiver. Control center PDC maintains for each PMU an array of recently received samples. Newly arrived synchrophasor is added, oldest synchrophasor is deleted from the array. Suppose, a PMU computes synchrophasors in random time sequence which is derived from equally spaced time instants t1, t2, t3….tt. The PMU has sent y1 ... y28, y30 synchrophasors corresponding to t1 … t28, t30 time instants. Now, PMU computes y33 and sends it to super-PDC. Once y33 reaches super-PDC, reconstruction process starts immediately to find y31, y32. It is to be noted that the reconstructed synchrophasors are equally spaced in time (t1, t2, ….tt). So, reconstructed synchrophasors can be easily used in the monitoring applications.
4.1.3
Modified SP Algorithm
In this work, original SP algorithm [42] is modified to reduce computational requirements. Modified SP algorithm reduces calculations for streaming synchrophasor data. Original SP algorithm [42] is an iterative process. Suppose, a signal is s-sparse or is expected to have maximum s non-zero elements in vector x. So, in every iteration of original SP [42], indices corresponding to s largest frequency components of signal are estimated. The s largest frequencies are then used to get an estimate of the original signal. The estimated signal is compared with the original signal to check the convergence of the algorithm.
In SP algorithm, pseudo inverse computations are the major computationally demanding steps. In this work, power system signals are expressed using Fourier/DCT basis. Structure of power system signals may change with time as power system goes through steady state and dynamic conditions. Three types of changes can happen in the co- efficient vector x of (2.24). In first type, magnitudes of frequency components may change; in second type, frequency pattern (positions of non-zero elements in x) may change; and in third type, both magnitude and frequency pattern may change. New pseudo inverse computation is required when coefficient pattern changes with respect to previous instant. However, new pseudo inverse computation is not needed if only the magnitudes of frequency components in vector x change with respect to previous instant. In the proposed modified SP algorithm, frequency pattern of previous data block is used as an initial estimate for the present data block. Frequency pattern of previous data block gives a good initial estimate for starting the iterative reconstruction process of present data block. Good initial estimate of frequency pattern leads to fewer iterations and faster convergence of the modified SP method. New pseudo inverse matrix computation is not needed when frequency pattern of present data block remains same as previous data block. As a result, computational burden gets reduced in the modified SP method. Many iterative power system algorithms use initial assumptions similar to modified SP method. In many power system dynamic or transient simulations, equations are initialized with values of previous time instant.
Modified SP algorithm differs from SP algorithm only at the initialization step. In step 1, frequency pattern Ti-1 of previous (i-1)th block of data is used as an input to the reconstruction algorithm of the ith block of data. In step 2, mismatch between original and estimated CS samples (using (i-1)th pattern) is computed. In step 3, algorithm stops if the norm of the mismatch vector is lower than a predefined value. The mathematical properties of modified SP are similar to the original SP method [42]. Results of this chapter prove the reduced computational requirement of modified SP method for synchrophasor data. The pseudo code of the proposed Modified Subspace pursuit algorithm is presented below.
Algorithm: Modified Subspace Pursuit Input:
A
,y
, Ti-1, s Initialization: 1) T0 = Ti-1 2)y
r0 y
A x
T0 T0; 3) If 0 2 ry , quit the iteration and output the estimated signal usingyx;
Else continue to step 4
Iteration: At the ℓth
iteration, go through the following steps
4) Tl Tl1 {s indices corresponding to the largest magnitude entries in the vector
* l 1
r
A y } *stands for matrix transposition 5) Set p †l T x A y where, † l T A is pseudo inverse of ATl 6) l
T = {s indices corresponding to the largest elements ofxpat ℓth iteration}
7) l l
l
r T T
y y A x ; where yrlis the residue vector at ℓth iteration
8) If 1
2 2,
l l
r r
Output:
9) The estimated coefficient vectorx, satisfying
x
{1,..., }N Tl0
and†
l l
T T
x A y
10) The estimated signal output is
y
x
where,
Ti-1 = S indices corresponding to the largest magnitude entries in the vector x for (i-1)th block of data.
T0 = Initial pattern of non-zero elements in x. 0
r
y = The initial residue (mismatch) vector.
0
T
x = Co-efficient vector estimated using T0 pattern/indices. 0
T
A = Sub-matrix of Awith columns matching T0 indices.
l
T
A = Sub-matrix of Awith columns matching Tl indices.
l r
y = Residue vector at ℓth iteration