阅读说明：本技术 电力系统同步相量测量方法 (Synchronous phasor measurement method for power system ) 是由 金涛 王晓岩 于 2020-10-20 设计创作，主要内容包括：本发明提供一种电力系统同步相量测量方法。由于电网的实际频率会出现偏移,导致传统的离散傅里叶变换在进行相量测量时会出现频谱泄露与栅栏现象,使得同步相量的测量结果出现误差。针对这一问题,本发明先推导出了三个相邻时刻的DFT数据之间的关系,利用此关系进行频率的跟踪,然后引入了极限学习机对非采样点进行求解,并且引入复化辛普森公式与梯形公式对DFT进行改进。在跟踪所得频率的基础上,将改进DFT测量的结果分为整数与分数部分,最后把两部分整合得出同步相量测量的结果。(The invention provides a synchronous phasor measurement method for a power system. Due to the fact that the actual frequency of the power grid can deviate, the phenomena of frequency spectrum leakage and barriers can occur when the traditional discrete Fourier transform is used for phasor measurement, and errors occur in measurement results of synchronous phasors. Aiming at the problem, the invention firstly deduces the relation among DFT data of three adjacent moments, tracks the frequency by using the relation, then introduces an extreme learning machine to solve non-sampling points, and introduces a complex Simpson formula and a trapezoidal formula to improve DFT. On the basis of tracking the obtained frequency, dividing the result of the improved DFT measurement into an integer part and a fractional part, and finally integrating the two parts to obtain the result of the synchronized phasor measurement.)

1. The method for measuring the synchronous phasor of the power system is characterized by comprising the following steps of:

acquiring the real-time frequency of a fundamental wave signal of the electric signal;

carrying out data fitting by using an extreme learning machine to estimate the non-sampling point;

and carrying out discretization processing on a result obtained by carrying out Fourier series transformation on the electric signal by combining a complex Simpson formula and a trapezoidal formula, and solving according to a numerical value obtained by estimation of an extreme learning machine to obtain the amplitude value and the phase angle value of the electric signal.

2. A power system synchrophasor measurement method according to claim 1, wherein said process of acquiring a real-time frequency of a fundamental signal of an electrical signal comprises the steps of:

calculating fundamental wave phasor values of fundamental wave signals at three adjacent moments;

establishing an equation based on the relationship between the fundamental wave phasor values at three adjacent moments;

and solving the established equation, and obtaining the actual frequency of the fundamental wave signal according to the solved solution.

3. The power system synchrophasor measurement method according to claim 2, wherein said process of performing data fitting using an extreme learning machine to estimate non-sampling points comprises the steps of:

performing least square calculation in the training set to obtain an optimal weight matrix between the hidden layer and the output layer;

putting the time points of the 1 st to the N +1 th sampling points and the sampling values into a training set, and fitting a function;

and (4) putting the time points of the non-sampling points into a training set, and solving the numerical values of the time points of the non-sampling points by the fitted function.

4. A method according to claim 3, wherein the step of obtaining the amplitude and phase angle values of the electrical signals comprises the steps of, in combination with the simpson's complex formula and the trapezoidal formula, performing discretization on the result obtained by performing fourier series transformation on the electrical signals, and performing solution according to the values estimated by the limit learning machine:

keeping the sampling frequency unchanged, and re-determining the actual sampling window length according to the real-time frequency of the fundamental wave signal;

dividing a result of Fourier series transformation on the continuous electric signal into two parts, and performing discretization processing on the two divided parts by combining a complex Simpson formula and a trapezoidal formula;

and substituting the numerical value estimated by the extreme learning machine into the result subjected to discretization processing to solve to obtain the amplitude and the phase angle value of the electric signal.

5. The power system synchrophasor measurement method according to claim 4, wherein said process of calculating fundamental phasor values of fundamental signals at three adjacent time instants comprises the steps of:

the fundamental wave signal of the sampled electrical signal is formula 1:

formula 1:

where a represents an effective value of the electric signal, ω ═ 2 π f ═ 2 π (f)₀+ Δ f), f is the actual frequency, f₀＝50Hz，Δf＝f-f₀，Is the initial phase angle of the electrical signal;

performing fourier series transformation on the continuous signal of formula 1 to obtain formula 2:

formula 2:

wherein, a₀Representing a direct current component, a_kAnd b_kIs the Fourier coefficient, M is the highest harmonic of Fourier decomposition, k is the harmonic order, omega₀Representing angular frequency, t being time;

transformation of the euler formula on equation 2 yields equation 3:

formula 3:

wherein k is the harmonic order, ω₀The angular frequency corresponding to the power frequency, j is a complex number mark, c_kIs the coefficient of the harmonic of the k-th order,c_kin the expression (a), T is a period;

discretizing the fundamental wave signal after Euler's formula conversion, and assuming the sampling frequency as f_c＝Nf₀Then, the sampling value of the sampling point at the nth time is represented by equation 4:

formula 4:

wherein n is 0,1, 2.;

equation 4 is converted to equation 5 by the euler equation:

formula 5:

wherein the content of the first and second substances,

connecting the vertical 3, the formula 4 and the formula 5 in parallel to obtain a formula 6:

formula 6:

wherein m is the number of the sampling points, the coefficient Pn is formula 7, the coefficient Qn is formula 8,the fundamental wave phasor value of the nth time point is obtained;

formula 7:

formula 8:

wherein the content of the first and second substances,

equations 9 and 10 are obtained from equations 7 and 8:

formula 9: p_n+1＝P_n·v；

Formula 10: q_n+1＝Q_n·v^-1；

Wherein

The DFT is used for solving fundamental wave phasor values at three moments of n, n +1 and n +2, as shown in formulas 11, 12 and 13:

formula 11:

formula 12:

formula 13:

6. the power system synchrophasor measurement method according to claim 5, wherein the equation established based on the relationship between the fundamental phasor values at three adjacent time instants is equation 14:

equation 14:

solving equation 14 yields the value of v, and solving for the actual frequency f of the fundamental signal by v, as shown in equation 15:

formula 15:

where Re (v) represents the real part of v and im (v) represents the imaginary part of v.

7. A synchronized phasor measurement method according to claim 6, wherein said process of dividing the result of said Fourier series transformation of said continuous electrical signal into two parts and discretizing said two parts by combining the Simpson's and trapezoidal equations comprises the steps of:

fourier series transformation is performed on the continuous electric signal y (t) to obtain the formula 27:

formula 27:

wherein, T is a period, T is time, j is a complex symbol, and omega is an angular frequency corresponding to an actual frequency;

the results of equation 27 are divided into two parts, equation 28, equation 29, equation 30:

formula 28:

formula 29:

formula 30:

wherein, T_cIs the sampling period, N 'is the re-determined actual sampling window length, G is the integer part of N';

when G +1 is an odd number, carrying out discretization calculation on the formula 29 by using a complex Simpson formula to obtain a formula 31:

formula 31:

wherein k is the serial number of the sampling point;

if G +1 is even, discretizing the formula 29 by matching the complex Simpson formula with the trapezoidal formula to obtain the formula 32:

formula 32:

discretization calculation is performed on equation 30 using a trapezoidal equation to obtain equation 33:

formula 33:

where y (N + N') is the non-sampled point.

8. A synchronous phasor measurement method for a power system according to claim 7, wherein the process of solving the value estimated by the extreme learning machine by substituting the value into the result after discretization to obtain the amplitude and phase angle values of the electrical signal comprises the following steps:

since the value y (N + N') estimated by the extreme learning machine is substituted into equation 33 to be solved, equation 28 is discretized to obtain equation 34:

formula 34: f₁(n)＝F_1i(n)+F_1f(n)；

Wherein, F₁(n) represents the phasor value at the nth time;

the amplitude A of the electric signal at the r-th time is obtained by using the equation 34_rAnd the phase angle valueAmplitude A_rAs shown in equation 35, the phase angle valueAs shown in equation 36:

formula 35:

formula 36:

wherein, Re [ F ]₁(n)]Is represented by F₁(n) real part, Im [ F ]₁(n)]Is represented by F₁(n) imaginary parts.

Technical Field

The invention belongs to the technical field of power system detection, and particularly relates to a synchronous phasor measurement method for a power system.

Background

Nowadays, the synchronous phasor measurement technology is widely applied to the fields of power system detection, protection and the like. If the accuracy of the synchronous phasor measurement method is improved, the reliability and safety of the power system can be further guaranteed, so that the improvement of the accuracy of the synchronous phasor measurement method is very important.

Discrete Fourier Transform (DFT) is widely used in the field of synchronized phasor measurement in power systems because of its simple principle, easy implementation, and certain harmonic suppression capability. When the power frequency is in a power frequency state, the measurement result obtained by using the DFT is very accurate, but if the DFT is used under the condition that the power frequency is floated, the asynchronous sampling problem occurs, and the phenomena of frequency spectrum leakage and barriers are generated, so that the measurement accuracy is reduced.

In view of this problem, those skilled in the art have proposed a method of introducing frequency tracking based on extended kalman filtering, and based on this, using DFT to perform synchrophasor measurement, which has the problems of huge calculation amount and poor speed. The person skilled in the art also proposes a symmetric translation sampling window method, which has good measurement accuracy and speed, but when the method is applied to practical situations, the virtual phasor deviates in the offset angle due to the fixed sampling frequency, and finally the dynamic error is not weakened obviously.

Disclosure of Invention

In order to solve the above technical problems, the present invention provides a method for measuring a synchronous phasor of a power system. The following presents a simplified summary in order to provide a basic understanding of some aspects of the disclosed embodiments. This summary is not an extensive overview and is intended to neither identify key/critical elements nor delineate the scope of such embodiments. Its sole purpose is to present some concepts in a simplified form as a prelude to the more detailed description that is presented later.

The invention adopts the following technical scheme:

in some alternative embodiments, it is to be collated.

The invention has the following beneficial effects: the accuracy of DFT is improved by combining the complex Simpson formula and the trapezoidal formula, so that the synchronous phasor measurement has higher accuracy; the accuracy of the fractional part of the DFT algorithm is improved by estimating the non-sampling points through the extreme learning machine; the dynamic characteristic of the synchronized phasor measurement is improved, and the synchronized phasor measurement has good dynamic adaptive adjustment capability to the emergency, so that the synchronized phasor measurement of the power system is realized, and effective support is provided for the reliability, safety and other aspects of the power system.

Drawings

FIG. 1 is a schematic flow diagram of the present invention;

FIG. 2 is a diagram illustrating DFT calculation errors under asynchronous sampling;

FIG. 3 is a diagram of an extreme learning machine;

FIG. 4 is a plot of non-sampled point predicted relative error at frequency offset.

Detailed Description

The following description and the drawings sufficiently illustrate specific embodiments of the invention to enable those skilled in the art to practice them. Other embodiments may incorporate structural, logical, electrical, process, and other changes. The examples merely typify possible variations. Individual components and functions are optional unless explicitly required, and the sequence of operations may vary. Portions and features of some embodiments may be included in or substituted for those of others.

As shown in fig. 1-4, in some illustrative embodiments, a power system synchronized phasor measurement method is provided, including the steps of:

firstly, acquiring an electric power signal, and performing frequency tracking by using a three-point method, namely acquiring the real-time frequency of a fundamental wave signal of the electric signal;

then, carrying out data fitting by using an extreme learning machine to estimate the non-sampling point;

and finally, performing synchronous phasor measurement by using the improved DFT algorithm of the composite Simpson formula and the trapezoidal formula, namely combining the composite Simpson formula and the trapezoidal formula, performing discretization processing on a result obtained by performing Fourier series transformation on the electric signal, and solving according to a numerical value estimated by an extreme learning machine to obtain the amplitude value and the phase angle value of the electric signal.

Frequency tracking by three-point methodBefore tracking, acquiring power signal, determining sampling frequency fc, and according to fc-Nf₀The initial window length N is determined, and the number of sampling points is determined to be N + 1.

Wherein the sampling frequency fc is determined according to the sampling frequency of the actual data acquisition device.

Since the actual frequency is not known at first, the power frequency f is first used₀The initial window length N is obtained, i.e. assuming the actual frequency at this time to be f₀Then, the initial window length N is used to track the actual frequency, and the result obtained by tracking is used to correct the window length N ', and the corrected window length N' is used to perform DFT synchronous phasor measurement.

In fig. 1, n-0 means that the following steps are performed to calculate the phasor value of the electrical signal at the time when n-0 is calculated, and then the phasor value of the electrical signal at the time when n-1 is calculated after the calculation is finished, and so on, that is, the loop in fig. 1 indicates that the phasor values of the electrical signals at all the time points are calculated one by one.

In some illustrative embodiments, the process of frequency tracking by the three-point method, i.e., the process of acquiring the real-time frequency of the fundamental signal of the electrical signal, comprises the steps of:

step one, calculating fundamental wave phasor values of fundamental wave signals at three adjacent moments:

the fundamental wave signal of the sampled electrical signal is assumed to be a cosine signal shown in formula 1:

formula 1:

where a represents an effective value of the electric signal, ω ═ 2 π f ═ 2 π (f)₀+ Δ f), f is the actual frequency, f₀＝50Hz，Δf＝f-f₀，Is the initial phase angle of the electrical signal.

Performing fourier series transformation on the continuous signal of formula 1 to obtain formula 2:

formula 2:

wherein, a₀Representing a direct current component, a_kAnd b_kIs the Fourier coefficient, M is the highest harmonic of Fourier decomposition, k is the harmonic order, omega₀Representing angular frequency, t being time.

Transformation of the euler formula on equation 2 yields equation 3:

formula 3:

wherein k is the harmonic order, ω₀The angular frequency corresponding to the power frequency, j is a complex number mark, c_kIs the coefficient of the harmonic of the k-th order,k＝1,2,3,...，c_kwherein T is a period.

Discretizing the fundamental wave signal after Euler's formula conversion, and assuming the sampling frequency as f_c＝Nf₀And N is the window length, the sampling value of the sampling point at the nth time is formula 4:

formula 4:

wherein n is 0,1, 2.

Equation 4 is converted to equation 5 by the euler equation:

formula 5:

wherein the content of the first and second substances,

connecting the vertical 3, the formula 4 and the formula 5 in parallel to obtain a formula 6:

formula 6:

wherein m is the number of the sampling points, the coefficient Pn is formula 7, the coefficient Qn is formula 8,is the fundamental phasor value at the nth time point.

Formula 7:

formula 8:

wherein the content of the first and second substances,

equations 9 and 10 are obtained from equations 7 and 8:

formula 9: p_n+1＝P_n·v；

Formula 10: q_n+1＝Q_n·v^-1；

Wherein the content of the first and second substances,

the DFT is used for solving fundamental wave phasor values at three moments of n, n +1 and n +2, as shown in formulas 11, 12 and 13:

formula 11:

formula 12:

formula 13:

secondly, establishing an equation based on the relationship between the fundamental wave phasor values at three adjacent moments: the equation established is equation 14, i.e. equation 14 can be obtained from the relationship among equations 11, 12, and 13:

equation 14:

the third step: solving the established equation, and obtaining the actual frequency of the fundamental wave signal according to the solved solution: solving equation 14 yields the value of v, and solves the exact actual frequency f of the fundamental signal by v, as equation 15:

formula 15:

where Re (v) represents the real part of v and im (v) represents the imaginary part of v.

In summary, frequency tracking is performed by using a three-point method, that is, phasor values at three adjacent moments are calculated by using a conventional DFT, an equation is established by using a mathematical relationship between the phasor values at the three adjacent moments, and finally, a solution of the equation is calculated to realize frequency tracking.

And tracking the frequency is realized, namely the number of sampling points can be adjusted according to the tracked frequency. The sampling frequency is the frequency at which the electrical signal data is sampled, and therefore the sampling frequency f_cIs fixed, sampling frequency f_cThe relation between the actual electric signal frequency f and the window length N satisfies f_cNf. The traditional method does not carry out frequency tracking and directly utilizes the formula f_c＝Nf₀Performing a calculation of where f₀Is the power frequency 50hz, the value of the window length N is fixed by the approximate calculation. The method tracks the frequency, namely tracks the actual frequency to obtain an estimated value f close to the actual frequency, wherein the f is changed along with the change of the actual frequency, and then the f is_cWhen f is changed, f is known as Nf_cFixed, the value of N will change with the change of f, so the number of sampling points can be adjusted according to the tracked frequency.

f_cNf means: for example, for sin function with frequency fLine discrete sampling, assuming f_cThen N +1 sample points are included in one period of the sin function. Since the N +1 sampling points are all in one period, the phasor value can be calculated more accurately when the N +1 sampling points are used for DFT calculation. If the frequency of the sin function becomes f at this time₁If f is still used_cThe number of N +1 is calculated as Nf, then N +1 sampling points will not all be in the same period, and then the phasor value error obtained when the DFT calculation is performed by using the N +1 sampling points will be very large. The invention therefore derives a varying value of N as the frequency of the sin function varies.

Since the subsequent use of the improved DFT requires the use of data at non-sampled times, the present invention estimates the non-sampled points by fitting the data using an extreme learning machine.

The extreme learning machine is a feedforward neural algorithm based on a single hidden layer, and weights between an input layer and the hidden layer and a threshold value of the hidden layer are randomly generated. Compared with the traditional neural network, the extreme learning machine greatly reduces the parameters to be set, so that the parameter searching time is greatly reduced, and the learning efficiency is improved.

In some illustrative embodiments, a process for estimating non-sampled points using data fitting with an extreme learning machine includes the steps of:

the first step is as follows: performing least square calculation in a training set to obtain an optimal weight matrix between a hidden layer and an output layer, putting the time points of 1 st to N +1 th sampling points and the sampling values into the training set, and fitting a function:

setting a time value t corresponding to each sampling point of a matrix input to the ELM model_sComposition, formula 16:

formula 16: t is_n×1＝[t₁ t₂ … t_n]^T _n×1；

Wherein n is the number of samples,l is a hidden layerAnd (4) the number of nodes.

The matrix of model prediction output values consists of the sampling prediction values of each sampling point, as shown in equation 17:

formula 17: c_n×1＝[c₁ c₂ … c_n]^T _n×1；

Wherein n is the number of samples, L is the number of nodes of the hidden layer, c₁To c_nFor predicted values of sampling points, e.g. c₁Predicting data for the electrical signal of the first sample point, c_nData is predicted for the electrical signal at the nth sample point.

The matrix of the true output values of the training samples consists of the true sample values of each sample point, as shown in equation 18:

formula 18: y is_n×1＝[y₁ y₂ … y_n]^T _n×1；

Wherein, y_nL is the number of nodes in the hidden layer.

The number of nodes of the hidden layer is set to be L, the weight between the input layer and the hidden layer is randomly generated by the system and is set to be omega_L×1, formula 19:

formula 19: omega_L×1＝[ω₁ ω₂ … ω_L]^T _L×1；

Wherein, ω is₁The number of input layer nodes of the ELM network constructed by the invention is only 1, omega_LIs the weight between the lth hidden layer node and the input layer node.

The threshold of the hidden layer is randomly generated by the system and is set as b_L×1As shown in formula 20:

formula 20: b_L×1＝[b₁ b₂ … b_L]_L×1 ^T；

Wherein, b₁Threshold for the first hidden layer node, b_LIs the threshold of the lth hidden layer node.

Output data H of the hidden layer_L×nFrom equation 21, it can be derived:

formula 21：

Wherein g (x) is an activation function, x₁For the time value, x, corresponding to the first sample point_nAnd the time value corresponding to the nth sampling point.

Weight matrix β between the hidden layer and the output layer:

unlike conventional learning algorithms, the ELM has not only minimal training errors, but also minimal output weight norms. According to the Bartlett theory, for the feedforward neural network with smaller training error, the smaller the norm of the weight is, the better the generalization performance is. The goal of ELM is to minimize the norm of the training error and output weights, as shown in equation 22:

formula 22: minimize | | | H β -Y | | | and | | | | β |;

the best β can be obtained by performing the least square calculation on equation 22 in the training set, as equation 23:

formula 23:

wherein the content of the first and second substances,referred to as the generalized inverse matrix, Y represents a matrix consisting of the actual values of the electrical signals corresponding to all the sampling points, see equation 18.

β_L×1Is the weight between the hidden layer and the output layer, as shown in equation 24:

formula 24: beta is a_L×1＝[β₁ β₂ … β_L]^T _L×1；

Wherein, beta₁The number of nodes of the input layer of the ELM network constructed by the invention is only 1, beta, for the weight between the first hidden layer and the output layer_LIs the weight between the lth hidden layer and the output layer.

The second step is that: putting the time points of the non-sampling points into a training set, solving the numerical values of the time points of the non-sampling points through the fitted function, and putting the parameters into a prediction set to obtain predicted values, wherein the formula is as follows, 25:

formula 25:

where h (x) is the output of the hidden layer.

In some illustrative embodiments, the process of obtaining the amplitude value and the phase angle value of the electrical signal by performing discretization on the result obtained by performing fourier series transformation on the electrical signal in combination with the complex simpson formula and the trapezoidal formula and solving the result according to the value estimated by the extreme learning machine includes:

in the first step, let the sampling frequency f_cAnd keeping the actual sampling window length N 'unchanged, and re-determining the actual sampling window length N' according to the actual frequency f obtained by tracking, i.e. the real-time frequency of the fundamental wave signal, as shown in formula 26:

formula 26:

wherein G is an integer part of N ', and G is a fractional part of N'.

The second step is that: dividing a result obtained by performing Fourier series transformation on a continuous electric signal into two parts, and performing discretization treatment on the two divided parts by combining a complex Simpson formula and a trapezoidal formula:

fourier series transformation is performed on the continuous electric signal y (t) to obtain the formula 27:

formula 27:

wherein, T is a period, T is time, j is a complex number mark, and ω is an angular frequency corresponding to the actual frequency.

The results of equation 27 are divided into two parts, equation 28, equation 29, equation 30:

formula 28:

formula 29:

formula 30:

wherein, T_cFor a sampling period, N 'is the re-determined actual sampling window length, and G is the integer part of N'.

When G +1, that is, the number of sampling points is odd, discretization calculation is performed on equation 29 by using only the complex simpson equation to obtain equation 31:

formula 31:

where k is the serial number of the sample point, e.g., the kth sample point.

If G +1, that is, the number of sampling points is even, discretizing calculation is performed on the formula 29 by matching the complex Simpson formula with the trapezoidal formula to obtain a formula 32:

formula 32:

the idea is that the parts from the 1 st sampling point to the G th sampling point are subjected to discretization calculation by a complex Simpson formula, and the parts from the G th sampling point to the G +1 th sampling point are subjected to discretization calculation by a trapezoid formula.

Discretization calculation is performed on equation 30 using a trapezoidal equation to obtain equation 33:

formula 33:

where g is the fractional part of N 'in equation 26 and y (N + N') is the non-sampled point.

The third step: substituting the numerical value estimated by the extreme learning machine into the result after discretization processing to solve to obtain the amplitude value and the phase angle value of the electric signal: since the value y (N + N') estimated by the extreme learning machine is substituted into equation 33 to be solved, equation 28 is discretized to obtain equation 34:

formula 34: f₁(n)＝F_1i(n)+F_1f(n)；

Wherein, F₁(n) represents the phasor value at the nth time;

the amplitude A of the electric signal at the r-th time is obtained by using the equation 34_rAnd the phase angle valueAmplitude A_rAs shown in equation 35, the phase angle valueAs shown in equation 36:

formula 35:

formula 36:

wherein, Re [ F ]₁(n)]Is represented by F₁(n) real part, Im [ F ]₁(n)]Is represented by F₁(n) imaginary parts.

Due to the fact that the actual frequency of the power grid can deviate, the phenomena of frequency spectrum leakage and barriers can occur when the traditional DFT carries out phasor measurement, and errors occur in measurement results of synchronous phasors. Aiming at the problem, the invention firstly deduces the relation among DFT data of three adjacent moments, tracks the frequency by using the relation, then introduces an extreme learning machine to solve non-sampling points, and introduces a complex Simpson formula and a trapezoidal formula to improve DFT. On the basis of tracking the obtained frequency, dividing the result of the improved DFT measurement into an integer part and a fractional part, and finally integrating the two parts to obtain the result of the synchronized phasor measurement.

The invention analyzes the relationship among DFT data of three adjacent time points under asynchronous sampling, deduces a frequency tracking formula by utilizing the relationship among the three data, adjusts the number of sampling points according to the tracking frequency, and then calculates by utilizing DFT optimized by a complex Simpson formula and a trapezoidal formula, wherein the calculation result is divided into two parts. The numerical value of the non-sampling point is used in the process, so the method of the ELM is provided, and the numerical value of the non-sampling point is estimated after the sampling point is trained.

Those of skill would further appreciate that the various illustrative logical blocks, modules, circuits, and algorithm steps described in connection with the embodiments disclosed herein may be implemented as electronic hardware, computer software, or combinations of both. To clearly illustrate this interchangeability of hardware and software, various illustrative components, blocks, modules, circuits, and steps have been described above generally in terms of their functionality. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the overall system. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present disclosure.