4.4.1
CS Sampling in PMU
Performance of CS reconstruction depends on the randomness of CS sampling. Random projection helps to retain signal information even at sub-Nyquist rate. PMU sampling algorithm should be designed to achieve higher reconstruction accuracies for various synchrophasors of C37.118.1-2011. In Table 4-21, effect of random sampling on synchrophasor reconstruction error (TVE) is presented. Matlab function ‘runstest’ is used to measure randomness of PMU sampling sequence. Function ‘runstest’ evaluates the hypothesis that the ‘sequence is random’. ‘Runstest’ returns the probability (p) of hypothesis ‘sequence is random’ being true. Lower value of probability doubts the validity of hypothesis ‘sequence is random’. Four different random sequences are considered in Table 4-21 for synchrophasor oscillations of 5 Hz. From Table 4-21, it is evident that the TVE errors decrease with increasing sampling randomness.
Table 4-21: Effect of random sampling on TVE
0.1,
x
k ka 0.1Xm 1.0
Probability (p)of Hypothesis ‘Sequence is Random’being True
TVE (%) Case-1 0.9 0.1978 0.8 0.2010 0.7 0.2252 0.6 0.4378
4.4.2
Data Window (N) and Sketch Length (m)
The computational requirement of CS reconstruction decreases as N decreases (the size of pseudo-inverse matrix reduces); but the reconstruction error increases. In this study, maximum TVE errors of Case 2 are always greater than Case 1. So, the choice of N is basically a design issue which can vary from case to case.
The value of m should be larger than the minimum value required by CS theory. It should also take into account the maximum possible numbers of missing synchrophasors during WAMS communication. Following equation can be used for choosing m. The value of l and FS should be chosen considering the type of communication network and the reconstruction algorithms.
_
mMinimum lengthl FS
where,
l= maximum possible missing data for a data window N FS= factor of safety
4.4.3
Sparsity (s)
Sparsity is an input parameter for CS algorithm. If actual sparsity of a signal is known priori, it can be used as an input during reconstruction. However, for many real world signals, only the maximum value of sparsity (maximum number of simultaneous frequency components) is known, rather than the actual sparsity. In this case, the maximum value of sparsity s is used as design input for CS. In CS literature, ‘maximum sparsity’ is often mentioned as ‘sparsity’.
In section 4.2, it is shown that synchrophasor measurements are sparse in nature. So, while designing CS for synchrophasor communication, the input value of sparsity (s) should be chosen considering maximum expected value of sparsity (s) in synchrophasor data. CS reconstruction error increases if actual sparsity of a signal is more than the chosen/design value of sparsity (s). The error limits of C37.118.1-2011 should be
satisfied with the chosen value of s. In this chapter, all previous results are computed considering maximum sparsity (s) equal to 4. In Tables 4-22 and 4-23, the effects of input sparsity values on the TVE are presented for oscillating and frequency ramp signals. In Table 4-22 results are presented for oscillating signals with input sparsity values 3 and 4. In Table 4-22, maximum TVE values are comparatively larger for input sparsity value 3 than sparsity value 4. Similarly, in Table 4-23 results are presented for frequency ramp signals with input sparsity values 3 and 4. In Table 4-23 also maximum TVE values are comparatively larger for input sparsity value 3 than sparsity value 4. The results of Table 4-22 and 4-23 imply that the actual sparsity values of investigated synchrophasor signals are greater than 3.
Table 4-22: Effect of sparsity on oscillating signals Modulation
Frequency (Hz)
Maximum TVE (%) for Case-1
s = 3 s = 4 0.1, x k ka 0.1Xm1.0 0.1 0.0038 0.0013 0.5 0.0203 0.0127 1 0.0414 0.019 2 0.0884 0.0612 3 0.1448 0.1116 4 0.1803 0.1791 5 0.3440 0.2171
Table 4-23: Effect of sparsity on frequency ramp signals Ramp Rate
(Hz/s)
Maximum TVE (%) computed over 2 s
s = 3 s = 4 Case 1 1 0.6174 0.3517 0.5 0.2912 0.1724 -0.5 0.2912 0.1724 -1 0.6174 0.3517
4.4.4
Bandwidth Savings
One of the purposes of CS is to reduce bandwidth requirement for synchrophasor communication. In this study, amount of bandwidth saving is expressed by N/m ratio. For non-sparse signals, TVE errors change with different values of m. Bandwidth savings should be designed in such a way that the TVE errors remain within the specified limits of [14] for all types of system conditions. In Table 4-24, the relations of bandwidth saving with maximum TVE values are presented for oscillating signals (kx=0.1, ka=0.1, modulation frequency is 10 Hz) considering window lengths N=48 and 52. The value of m is varied for each N to get different amounts of bandwidth savings. In Table 4-24, for window length N=48, TVE values remain almost same for the bandwidth savings between 3.0 to 3.69 times. Maximum TVE value increases drastically to 5.3199% as bandwidth saving reaches 4.00. In Table 4-24, results are also presented for window length N=52. In this case, maximum TVE remains within 1% for bandwidth savings up to 5.77. Maximum TVE increases drastically to 66.589% as bandwidth saving reaches 6.5.
The results of Table 4-24 signify that the amount of bandwidth saving is a design issue in CS and it depends on the value of N. Higher bandwidth savings may be achieved if the window length in increased. But, response time increases with increasing window length.
Table 4-24: Relation of bandwidth savings with TVE for oscillating synchrophasors Bandwidth Savings N=48 N=52 TVE (%) Response (in Fundamental Cycle) TVE (%) Response
(in Fundamental Cycle)
3.00 0.4854 1 - - 3.20 0.4813 1 - - 3.43 0.4797 1 - - 3.69 0.5590 1 - - 4.00 5.3199 1 0.7298 1.083 4.33 - - 0.7228 1.083 4.72 - - 0.7687 1.083 5.20 - - 0.5759 1.083 5.77 - - 0.8081 1.083 6.50 - - 66.589 1.083
4.4.5
Program Runtime
In this work, Matlab software, running on a computer with ‘Windows 7’ operating system, Intel Core i7 CPU and 4 GB RAM, has been used for all simulations. The maximum run time is less than 0.001 second for CS reconstructions. ‘Windows 7’ is a non-real time operating system. It is expected that the program run time may be reduced further by optimizing coding on a real time operating system.
High performance computers are usually used in PDCs. Several performance enhancing techniques such as separate process, dedicated processor, parallel computing, etc can be used to reduce the computational latency of PDCs. Recently, cloud computing platforms with massive computational abilities have been proposed for future smart grid computations. The computational latency of the proposed CS algorithm can be considered very small in future perspective.