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diff --git a/ma/safety_reset.tex b/ma/safety_reset.tex index 3462740..303b519 100644 --- a/ma/safety_reset.tex +++ b/ma/safety_reset.tex @@ -1011,30 +1011,45 @@ controllable load: \item[Modulation amplitude] proportionally related to modulation power. In a practical setup we might realize a modulation power up to a few hundred \si{\mega\watt} which would yield maybe a few tens of \si{\milli\hertz} of frequency amplitude. - \item[Modulation pre-emphasis and slew-rate control]. Pre-emphasis might be necessary to ensure an adequate SNR at - the receiver. Slew-rate control and other shaping measures might be necessary to reduce the impact of these - sudden load changes on the transmitter's primary function (say, aluminium smelting) and to prevent disturbances - to grid components. + \item[Modulation pre-emphasis and slew-rate control]. Pre-emphasis might be necessary to ensure an adequate + Signal-to-Noise ratio (SNR) at the receiver. Slew-rate control and other shaping measures might be necessary to + reduce the impact of these sudden load changes on the transmitter's primary function (say, aluminium smelting) + and to prevent disturbances to grid components. \item[Modulation frequency]. For a practical implementation a careful study would be necessary to determine an optimal frequency band for operation. On one hand we need to prevent disturbances to the grid such as through - excitation of some local or inter-area modes. On the other hand we need to optimize SNR and data rate to achieve - optimal latency between transmission start and successful reception and to reduce the overall burden on - transmitter and grid. + excitation of some local or inter-area modes. On the other hand we need to optimize Signal-to-Noise ratio (SNR) + and data rate to achieve optimal latency between transmission start and successful reception and to reduce the + overall burden on transmitter and grid. \item[Further modulation parameters]. The modulation itself has numerous parameters that are discussed in sec.\ \ref{mod_params} below. \end{description} -\section{From grid frequency to a reliable communications channel} -% FIXME +\section{From grid frequency to a reliable communication channel} \subsection{Channel properties} -% FIXME +In this section we will explore how we can construct a reliable communication channel from the analog primitive we +outline in the previous section. Our load control approach to grid frequency modulation leads to a channel with the +following properties. +\begin{description} + \item[Slow-changing.] Accurate grid frequency measurements need several periods of the mains sine wave. Faster + sampling rates can be achieved with more complex specialized synchrophasor estimation algorithms but this will + result in a tradeoff between sampling rate and accuracy\cite{belega01}. + \item[Analog.] Grid frequency is an analog signal. + \item[Noisy.] While stable over long periods of time thanks to Load-Frequency Control\cite{entsoe04} it shows + considerable random short-term variations. In addition our modulation amplitude is limited by technical and + economic constraints so we have to find a system that will work at poor SNRs. + \item[Polarized.] Grid frequency measurements have an inherent sense of \emph{up} (higher frequencies). We can use + this in a polarized modulation scheme to encode information without first transmitting some reference signal to + establish this polarization. +\end{description} \subsection{Modulation and its parameters} - \label{mod_params} +In this section we will consider how to select a good set of parameters for a modulation scheme fitting grid frequency +modulation. + The sensitivity of the grid to oscillation at particular frequencies described above means we should avoid any modulation technique that would concentrate a lot of energy in a small bandwidth. Taking this principle to its extreme provides us with a useful pointer towards techniques that might work well: Spread-spectrum techniques. By employing @@ -1285,12 +1300,12 @@ required precision for manageable averaging times--we would need either a ADC sa for a reconstruction through interpolated readings an impractically high ADC resolution. Detail on the inner workings of commercial phasor measurement units is scarce but given their essential role to SCADA -systems there is a large amount of academic research on such algorithms\cite{narduzzi01,derviskadic01}. A popular -approach to these systems is to perform a Short-Time Fourier Transform (STFT) on ADC data sampled at high sampling rate -(e.g. \SI{10}{\kilo\hertz}) and then perform some analysis on the frequency-domain data to precisely locate the strong peak -around \SI{50}{\hertz}. A key observation here is that FFT bin size is going to be much larger than required frequency -resolution. This fundamental limitiation follows from the nyquist criterion %FIXME maybe cite? and if we had to process -an \emph{arbitrary} signal this would highly limit our practical measurement accuracy +systems there is a large amount of academic research on such algorithms\cite{narduzzi01,derviskadic01,belega01}. A +popular approach to these systems is to perform a Short-Time Fourier Transform (STFT) on ADC data sampled at high +sampling rate (e.g. \SI{10}{\kilo\hertz}) and then perform some analysis on the frequency-domain data to precisely +locate the strong peak around \SI{50}{\hertz}. A key observation here is that FFT bin size is going to be much larger +than required frequency resolution. This fundamental limitiation follows from the nyquist criterion %FIXME maybe cite? +and if we had to process an \emph{arbitrary} signal this would highly limit our practical measurement accuracy \footnote{ Some software packages providing FFT or STFT primitives such as scipy\cite{virtanen01} allow the user to super-sample FFT output by specifying an FFT width larger than input data length, padding the input data with zeros @@ -1351,12 +1366,22 @@ resolution and despite numerous distortions. Published grid frequency estimation algorithms such as \textcite{narduzzi01} or \textcite{derviskadic01} are rather sophisticated and use a combination of techniques to reduce numerical errors in FFT calculation and peak fitting. Given that we do not need reference standard-grade accuracy for our application we chose to start with a very basic algorithm -instead. We chose to use a general approach developed by experimental physicists at CERN that is described by -\textcite{gasior01}. This approach assumes a general sinusoidal signal superimposed with harmonics and broadband noise. -Applicable to a wide spectrum of practical signal analysis tasks it is a reasonable first-degree approximation of the -much more sophisticated estimation algorithms developed specifically for power systems. Some algorithms have components -such as kalman filters\cite{narduzzi01} that require a phyiscal model. As a general algorithm from \textcite{gasior01} -does not require this kind of application-specific tuning, eliminating one source of error. +instead. We chose to use a general approach to estimate the precise fundamental frequency of an arbitrary signal that +was developed by experimental physicists at CERN and that is described by \textcite{gasior01}. This approach assumes a +general sinusoidal signal superimposed with harmonics and broadband noise. Applicable to a wide spectrum of practical +signal analysis tasks it is a reasonable first-degree approximation of the much more sophisticated estimation algorithms +developed specifically for power systems. Some algorithms have components such as kalman filters\cite{narduzzi01} that +require a phyiscal model. As a general algorithm from \textcite{gasior01} does not require this kind of +application-specific tuning, eliminating one source of error. + +The \textcite{gasior01} algorithm passes the windowed input signal through a DFT, then interpolates the signal's +fundamental frequency by fitting a wavelet such as a gaussian to the largest peak in the DFT results. The bias parameter +of this curve fit is an accurate estimation of the signal's fundamental frequency. This algorithm is similar to the +simpler interpolated DFT algorithm used as a reference in much of the synchrophasor estimation +literature\cite{borkowski01}. The three-term variant of the maximum sidelobe decay window often used there is a blackman +window with parameter $\alpha = \frac{1}{4}$. Analysis has shown\cite{belega01} that the interpolated DFT algorithm is +worse than algorithms involving more complex models under some conditions but that there is \emph{no free lunch} meaning +that more complex perform worse when the input signal deviates from their models. \subsection{Frequency sensor hardware design} \label{sec-fsensor} |