Harmonic detection method for power system

Abstract

The invention relates to the technical field of harmonic detection of a power system, in particular to a harmonic detection method combining dynamic window function optimization, fast Fourier transform (FFT), fractional Fourier transform (FRFT) and least square parameter correction, and is suitable for real-time monitoring and analysis of unsteady-state power signals. The method is especially suitable for inter-harmonic, sub-harmonic and frequency fluctuation problems in complex scenes such as new energy grid connection and industrial frequency conversion equipment. Through a hybrid algorithm, dynamic window optimization and a dual-core hardware architecture, the contradiction of a traditional method in precision, real-time performance and adaptability is solved, and the method is suitable for harmonic monitoring and treatment of an intelligent power grid, an industrial production line and a new energy system and has remarkable technical advancement and market application potential.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of power system detection, in particular to a harmonic detection method for a power system. BACKGROUND

[0002] Traditional power system harmonic detection mainly relies on fast Fourier transform (FFT) and its improved algorithm. However, FFT has inherent defects such as spectrum leakage, fence effect and insufficient frequency resolution in non-steady state signal analysis. Especially when the grid frequency fluctuates or there are interharmonics, the traditional method is difficult to accurately extract harmonic parameters. In addition, the existing technology mainly uses static window function or single algorithm optimization, which is difficult to balance real-time performance and accuracy. Moreover, the traditional single CPU architecture cannot fully utilize the parallel computing capability, resulting in low algorithm execution efficiency and waste of hardware resources.

[0003] Chinese patent CN111025015A discloses a harmonic detection method. This method can effectively improve the detection accuracy of the algorithm by using cosine combination window function and fast Fourier transform, and the spectrum leakage in short and long ranges is well suppressed. However, there is still room for improvement in the parameter optimization of the cosine combination window function and the introduction of more advanced signal processing and analysis methods.

[0004] In summary, the existing harmonic detection technology has the contradiction between accuracy and real-time performance, and a single algorithm cannot meet the requirements of high resolution and fast calculation in complex signal environment. In addition, the existing static window function and fixed parameter correction method cannot adapt to the time-varying characteristics of non-steady state signals, resulting in insufficient harmonic dynamic adaptability. Furthermore, the traditional single CPU architecture cannot fully utilize the parallel computing capability, resulting in low algorithm execution efficiency and waste of hardware resources. SUMMARY

[0005] The present application provides a harmonic detection method for a power system to solve the technical problems of the existing harmonic detection technology, such as the contradiction between accuracy and real-time performance, and the difficulty of a single algorithm to meet the requirements of high resolution and fast calculation in complex signal environment.

[0006] The present application uses a dual-core processor, combines dynamic window function optimization and hybrid algorithm design, improves the accuracy and speed of harmonic detection, reduces errors while meeting real-time requirements. The technical scheme includes the following steps:

[0007] Step 1: synchronously sampling the power grid signal, the power grid signal including voltage signal and / or current signal;

[0008] Step 2: anti-aliasing filtering the signal sampled in step 1;

[0009] Step 3: dynamically segmenting and windowing the signal filtered in step 2;

[0010] Step 4: The main CPU performs the initial spectrum extraction on the segmented and windowed signal of Step 3, specifically including: FFT (Fast Fourier Transform) calculation, harmonic coarse extraction, and spectrum leakage compensation;

[0011] Step 5: The slave CPU performs the inter-harmonic extraction on the segmented and windowed signal of Step 3, specifically including: FRFT (Fractional Fourier Transform) calculation, fractional order optimization, and inter-harmonic detection;

[0012] Step 6: Perform least squares parameter correction on the results of Step 4 FFT calculation and Step 5 FRFT calculation, optimize the parameters extracted in Step 4 and Step 5, specifically including: harmonic amplitude, frequency and phase, and inter-harmonic amplitude, frequency and phase;

[0013] Step 7: Integrate the calculation results of Steps 4, 5, and 6 to complete the total harmonic distortion calculation;

[0014] Step 8: Output signal.

[0015] Further, the anti-aliasing filter in Step 2 adopts a sixth-order Butterworth low-pass filter, applies bilinear transformation, and finally calculates the filtered signal.

[0016] Further, the signal dynamic segmentation and windowing in Step 3 specifically includes: judging whether the filtered signal of Step 2 is a stationary signal, if so, selecting Hanning window for windowing segmentation; if not, selecting Blackman window for windowing segmentation.

[0017] Further, the threshold of the stationarity index of the filtered signal of Step 2 is updated in real time, and the threshold is dynamically corrected according to historical data or environmental changes.

[0018] Further, a dual-core processor is adopted, and the main and slave CPUs process the segmented and windowed signal simultaneously; the main CPU performs FFT calculation, harmonic coarse extraction, and spectrum leakage compensation; the slave CPU performs FRFT calculation, fractional order optimization, inter-harmonic detection, and least squares parameter correction.

[0019] Further, in Step 4, the main CPU performs FFT calculation, harmonic coarse extraction, and spectrum leakage compensation on the windowed signal, specifically including: the main CPU reads the segmented and windowed signal; the main CPU performs FFT calculation on the read signal; the main CPU performs harmonic coarse extraction on the FFT calculation result; the main CPU performs spectrum leakage compensation on the coarse extracted harmonics, and the compensation method adopts the dual-spectrum line interpolation method.

[0020] Further, in step 5, the CPU performs fractional Fourier transform calculation, fractional order optimization, inter-harmonic detection and least square method combined optimization of FFT and FRFT parameters on the segmented and windowed signal; specifically including: reading the segmented and windowed signal from the CPU; performing FRFT calculation on the read signal from the CPU; performing fractional order optimization on the FRFT calculation result from the CPU, and the optimization target is to maximize the energy concentration of the inter-harmonic component; performing inter-harmonic detection on the optimized result from the CPU.

[0021] Further, in step 6, according to the output of FFT and FRFT, the least square parameter correction is performed to optimize the parameters of the initial spectrum analysis and inter-harmonic extraction; specifically including: performing least square parameter correction on the output of step 4 FFT and step 5 FRFT, and the correction target is to minimize the residual sum of squares between the FFT and FRFT results; performing amplitude optimization on the corrected parameters; performing frequency optimization on the optimized amplitude; performing phase optimization on the optimized frequency.

[0022] Technical features of the present application: (1) mixed algorithm design: FFT and FRFT work together, FFT quickly extracts fundamental and harmonic components, FRFT enhances non-integer harmonic detection, and comprehensively improves accuracy and speed.

[0023] (2) Dynamic window function optimization: based on the signal spectrum characteristics, Hanning window, Blackman window and the like are adaptively selected, and the least square method is combined to dynamically correct the harmonic parameters, thereby reducing the amplitude error and phase error.

[0024] (3) Introducing a least square parameter correction link to optimize the detected harmonic parameters, thereby improving the measurement accuracy of amplitude, frequency and phase.

[0025] (4) Anti-aliasing filter, dynamic signal segmentation and windowing preprocessing means effectively suppress the spectral leakage and fence effect, further improving the detection accuracy.

[0026] (5) Hardware acceleration architecture: dual-core processor division processing, main CPU performs FFT, and slave CPU completes FRFT and parameter correction, meeting the real-time requirement.

[0027] In summary, through the mixed algorithm, dynamic window optimization and dual-core hardware architecture, the present application solves the contradiction between the traditional method in accuracy, real-time and adaptability, and is suitable for harmonic monitoring and management of smart grid, industrial production line and new energy system, and has significant technical advancement and market application potential. BRIEF DESCRIPTION OF DRAWINGS

[0028] Figure 1 Harmonic detection algorithm flowchart of the present application. DETAILED DESCRIPTION

[0029] This invention relates to the field of power system harmonic detection technology, specifically to a harmonic detection method that combines dynamic window function optimization, fast Fourier transform (FFT), fractional Fourier transform (FRFT), and least squares parameter correction. It is applicable to real-time monitoring and analysis of non-steady-state power signals, especially for interharmonic, subharmonic, and frequency fluctuation problems in complex scenarios such as new energy grid connection and industrial frequency conversion equipment.

[0030] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0031] This embodiment is an inventive method implemented in power system harmonic detection. The same method is also applicable to harmonic monitoring and control in smart grids, industrial production lines, and new energy systems. Figure 1 This is a schematic diagram of the algorithm designed for this invention.

[0032] like Figure 1 As shown, this invention first synchronously samples the power grid signal (voltage / current), then performs anti-aliasing filtering on the sampled signal, next determines the stability of the filtered signal and dynamically selects a window function to complete the segmented windowing of the signal. Then, the main CPU and slave CPU simultaneously calculate the segmented windowed signal. The main CPU performs FFT (Fast Fourier Transform) calculation, harmonic coarse extraction, and spectral leakage compensation on the segmented windowed signal, while the slave CPU performs FRFT (Fractional Fourier Transform) calculation, fractional-order optimization, and interharmonic detection. Subsequently, based on the results calculated by the main and slave CPUs, the least squares method is used to simultaneously optimize the parameters extracted by FFT and FRFT. Finally, the calculation results are integrated to complete the calculation, display, and transmission of the harmonic signal.

[0033] I. Signal Synchronous Acquisition Design:

[0034] Synchronous sampling of power grid signals specifically includes:

[0035] An analog-to-digital converter (ADC) is used to convert the mains voltage or current signal from analog to digital to obtain a digital signal. The obtained digital signal is then synchronously sampled at a sampling frequency of f. s The sampling period is T s The power grid operates at frequency f1. The sampled digital signal is stored in a buffer.

[0036] The analog signal input to the power grid is:

[0037] It includes fundamental frequency, harmonics, and noise.

[0038] Where h is the harmonic order, K is a positive integer, and A h f is the amplitude of the h-th harmonic. h f is the frequency of the h-th harmonic.h = h - f1, φ h is the phase of the h-th harmonic, and ε(t) is the noise signal.

[0039] The discretized signal is:

[0040] x(n) = υ(nT s )

[0041] where n is an integer, n = 0, 1, 2, 3,..., representing the nth sampling point, T s is the sampling interval, T s = 1 / f s , f s is the sampling frequency.

[0042] The output signal after synchronous acquisition is:

[0043] II. Anti-aliasing filter design:

[0044] The sampled signal is processed by an anti-aliasing filter, which includes:

[0045] The stored digital signal is read from the buffer area, and the read digital signal is processed by a digital filter to filter out high-frequency components above the cutoff frequency. In this embodiment, the digital filter is a Butterworth low-pass filter, which is applied to bilinear transformation, and finally the filtered signal is calculated. The cutoff frequency of the Butterworth low-pass filter is f c , The order is 6.

[0046] The transfer function of the 6th order Butterworth low-pass filter is decomposed into a cascade form of three 2nd order sections:

[0047]

[0048] s is a complex frequency variable, s = jω, j is the imaginary unit, and ω = 2πf is the angular frequency.

[0049] k is a loop variable, k = 1, 2, 3.

[0050] After applying bilinear transformation, the filtered difference equation is:

[0051]

[0052] n is the discrete time, representing the current time, x(n) is the current input, and y(n) is the current output.

[0053] i is the delay, representing the number of delay steps relative to the current time n.

[0054] b iare the forward filter coefficients, which determine the weight of the input signal x(n) and its history values on the current output, totally 7, i=0, 1, 2, 3, 4, 5, 6.

[0055] a i are the feedback filter coefficients, which determine the weight of the output signal y(n) and its history values on the current output, totally 6, i=1, 2, 3, 4, 5, 6.

[0056] The formula shows that at the n th moment, the latest collected input sample x(n) and the stored past 6 input samples x(n-1)...x(n-6), past 6 output samples y(n-1)...y(n-6) are weighted and summed, the coefficients are a i , b i , so as to calculate the current new output value y(n).

[0057] b0, b1,..., b6 and a1, a2,..., a6 are calculated by bilinear transformation.

[0058] III. Signal segmentation and window design:

[0059] Determine whether the filtered signal is a stationary signal, if yes, select Hanning window for signal window segmentation; otherwise, select Blackman window for signal window segmentation, which specifically includes:

[0060] Use the stationarity index to determine whether the filtered signal is a stationary signal. The stationarity index is Q, and the threshold of the stationarity index is Q th , if the stationarity index Q th is less than the threshold Q th , select Hanning window for signal window segmentation; if the stationarity index Q is greater than or equal to the threshold Q th , select Blackman window for signal window segmentation. The segmentation length is (i.e. window length) N, and the overlap rate is 50% (i.e. the step length is N / 2).

[0061] The stationarity index formula is as follows:

[0062]

[0063] Wherein, μ is the mean value of the signal segment.

[0064] Q th is updated in real time, and the threshold is dynamically corrected according to the historical data or environmental changes, and the formula is:

[0065] Q th (t) = αQ th (t-1) + (1-α)·Q new

[0066] where a is the forgetting factor, usually 0.9≤a<1, Q new is the latest statistical value.

[0067] The Hann window function formula is (used for stationary signal segmentation windowing):

[0068]

[0069] The Blackman window function formula is (used for non-stationary signal segmentation windowing):

[0070]

[0071] Thus, the mth segment of the windowed signal is:

[0072]

[0073] where N represents the size of the window, which is a fixed value, n represents the position in the window, which is a variable, and n is an integer less than N.

[0074] III. Main CPU algorithm design:

[0075] The main CPU performs fast Fourier transform calculation, harmonic rough extraction and spectrum leakage compensation on the windowed signal. Specifically, it includes:

[0076] 1. The main CPU reads the segmented and windowed signal.

[0077] 2. The main CPU performs FFT calculation on the read signal, with the FFT point number being N / 2 points. Specifically as follows:

[0078] The FFT calculation is performed on the signal x m obtained after segmentation and windowing, to obtain the following spectrum:

[0079]

[0080] where k is an integer variable representing the kth spectrum line in the spectrum obtained after FFT calculation, 0≤k k < N, and j is the imaginary unit. Each spectrum line k corresponds to a specific frequency f

[0081]

[0082] f s is the sampling rate, and N is the FFT point number.

[0083] Harmonic frequency point positioning is the process of accurately determining the frequency of each harmonic in the signal through FFT.

[0084] Harmonic frequency point positioning is as follows:

[0085]

[0086] k h f represents the theoretical frequency f of the h-th harmonic in the FFT spectrum. h The corresponding nearest frequency is an integer calculated for a specific harmonic.

[0087] 3. The main CPU performs coarse harmonic extraction on the FFT calculation results. Details are as follows:

[0088] The spectrum signal X obtained by the above FFT m Harmonic coarse extraction is performed to locate the fundamental frequency as follows:

[0089] k1 is the position of the fundamental frequency in the FFT spectrum.

[0090] The harmonic search range is:

[0091] k h ∈[round(hk1-Δ,round(hk1+Δ)]

[0092] Where, k h Let h be the frequency of the h-th harmonic, h be the harmonic order, and Δ be the radius of the search window, which is used to expand the possible location range of the harmonics and avoid frequency deviation caused by spectrum leakage or noise.

[0093] 4. The main CPU performs spectral leakage compensation on the coarsely extracted harmonics using a dual-spectral-line interpolation method. Details are as follows:

[0094] The formula for double spectral line interpolation is as follows:

[0095]

[0096] Where δ is the frequency offset correction factor, used to estimate the deviation between the actual harmonic frequency and the discrete FFT frequency point.

[0097] The corrected frequency, amplitude, and phase are as follows:

[0098]

[0099] IV. From the perspective of CPU algorithm design:

[0100] The CPU performs fractional Fourier transform calculations, fractional-order optimization, interharmonic detection, and least-squares joint optimization of FFT and FRFT parameters on the segmented, windowed signal. Specifically, this includes:

[0101] 1. Read the segmented and windowed signals from the CPU;

[0102] 2. Perform FRFT calculation on the signal read from the CPU. Details are as follows:

[0103] The fractional Fourier transform is:

[0104]

[0105] The kernel function K p The expression is:

[0106]

[0107] Where α = pπ / 2, p is the fractional order (0≤p≤2), and u is the fractional domain variable corresponding to the rotated time-frequency plane coordinate.

[0108] After the discrete FRFT processing, the signal expression is:

[0109]

[0110] 3. From the CPU, the fractional order of the FRFT calculation result is optimized, and the optimization goal is to maximize the energy concentration degree of the inter-harmonic component. The specific steps are as follows:

[0111] Select the optimal order P opt , so that in the FRFT domain, the signal energy concentration degree is maximum, and the spectrum leakage is minimum.

[0112] According to the FRFT domain signal X p (u) and the FFT modified f h , the gradient descent iteration method is used to calculate the optimal order P opt is as follows:

[0113]

[0114] Where, u h represents the center coordinate of the hth harmonic component in the fractional FRFT domain u.

[0115] U h represents the harmonic frequency band range in the FRFT domain, U h is an interval, defined as the neighborhood of 2Δu centered at u h .

[0116]

[0117] Where, Δu is the bandwidth tolerance.

[0118] 4. From the CPU, the inter-harmonic detection is performed on the optimized result. The specific steps are as follows:

[0119] The optimized FRFT domain signal is calculated as:

[0120] X Popt (u) = FRFT(x m(n), P opt )

[0121] In the FRFT domain, search for non-integer multiple frequency points, the extracted frequency is:

[0122]

[0123] In order to avoid confusion, ensure that the inter-harmonic frequency is at least 0.5Hz apart from the fundamental / harmonic frequency, the detection condition is as follows:

[0124]

[0125] The extracted amplitude is:

[0126] A inter,m = |X Popt (u peak )|

[0127] The extracted phase is:

[0128]

[0129] The expression of inter-harmonic is:

[0130]

[0131] 5、According to the output of FFT and FRFT, the least square parameter correction is carried out, and the parameters of initial spectrum analysis and inter-harmonic extraction are optimized. Specifically, it includes:

[0132] 5.1 The least square parameter correction is carried out on the output of step FFT and FRFT, and the correction target is to minimize the residual sum of squares between FFT and FRFT results;

[0133] 5.2 The amplitude optimization is carried out on the corrected parameters;

[0134] 5.3 The frequency optimization is carried out on the optimized amplitude;

[0135] 5.4 The phase optimization is carried out on the optimized frequency.

[0136] The fundamental and harmonic parameters extracted by FFT: amplitude A h , frequency f h , phase Φ h

[0137] The inter-harmonic parameters detected by FRFT: amplitude A inter , frequency f inter , phase Φ inter .

[0138] The harmonic time domain model composed of FFT and FRFT is:

[0139]

[0140] K is the total number of harmonics extracted by FFT, and M is the total number of inter-harmonics extracted by FRFT.

[0141] The residual sum of squares of the minimum model and the measured signal is:

[0142]

[0143] where y(n) is the filtered measured discrete signal.

[0144] The least square method is used to optimize the parameter vector:

[0145]

[0146] The Jacobian matrix H is constructed, and each column of the Jacobian matrix H is the partial derivative of the residual with respect to the parameter:

[0147]

[0148] where the residual r(n) = y(n) - υ(nT s , and the specific partial derivative is:

[0149] The partial derivative of the harmonic amplitude A h :

[0150] The partial derivative of the harmonic frequency f h :

[0151] The partial derivative of the harmonic phase Φ h :

[0152] The partial derivative of the inter-harmonic amplitude A inter :

[0153] The partial derivative of the inter-harmonic frequency f inter :

[0154] The partial derivative of the inter-harmonic phase Φ inter :

[0155] The parameter vector θ is updated by iteration:

[0156] θ (k+1) = θ (k) -l(Η T Η+λΙ) -1 Η T r

[0157] wherein, l is the step length, λ is the damping factor, r = [r(0), r(1),..., r(N-1)] T is the residual vector. The iteration termination condition is:

[0158] Residual change: ||r (k+1) (k) || < ε, maximum iteration number k max = 50.

[0159] The final corrected objective function is:

[0160]

[0161] wherein, y(n) is the filtered measured discrete signal.

[0162] Five, data integration:

[0163] The calculation results are integrated to complete the calculation of harmonics and the drawing of the spectrum. Specifically, it includes:

[0164] According to the calculated and corrected FFT harmonic parameters {A h , f h}, FRFT interharmonic parameters {A inter , f inter}, the final harmonic content can be calculated as:

[0165]

[0166] wherein, K represents the highest harmonic order, A1 represents the fundamental amplitude, A h h represents the amplitude of the hth harmonic, A inter represents the interharmonic amplitude, and M represents the number of interharmonics.

[0167] Six, data output:

[0168] The harmonic signal is output, including the amplitude, frequency and phase of each order of harmonic, specifically including:

[0169] Harmonic total content THD;

[0170] FFT extracted fundamental and harmonic parameters: amplitude A h , frequency f h , phase Φ h ;

[0171] FRFT detected interharmonic parameters: amplitude A inter , frequency f inter , phase Φ inter .

[0172] ​Finally, through Modbus, MQTT, etc. protocol upload to the cloud or SCADA system, while the data stored in the Flash cycle.

Claims

1. A method for harmonic detection for power systems, characterized by: The method comprises the following steps: Step 1: synchronously sampling a power grid signal, the power grid signal comprising a voltage signal or a current signal; Step 2: performing anti-aliasing filtering on the signal sampled in step 1; Step 3: performing dynamic segmentation and windowing on the signal filtered in step 2; Step 4: performing primary spectrum extraction on the signal segmented and windowed in step 3 by a main CPU, specifically comprising: fast Fourier transform calculation, harmonic coarse extraction and spectrum leakage compensation; Step 5: performing inter-harmonic extraction on the signal segmented and windowed in step 3 by a slave CPU, specifically comprising: fractional Fourier transform calculation, fractional order optimization and inter-harmonic detection; Step 6: performing least square parameter correction on the results of the fast Fourier transform calculation in step 4 and the fractional Fourier transform calculation in step 5, optimizing the parameters extracted in steps 4 and 5, and the specific optimized parameters comprising: the amplitude, frequency and phase of the harmonics and the amplitude, frequency and phase of the inter-harmonics; Step 7: performing data integration on the calculation results in steps 4, 5 and 6, and completing total harmonic distortion calculation; Step 8: outputting a signal.

2. The method for harmonic detection for power systems as claimed in claim 1, wherein, The anti-aliasing filter in step 2 adopts a sixth-order Butterworth low-pass filter, and a bilinear transformation is applied to finally calculate the filtered signal.

3. The method for harmonic detection for power systems as claimed in claim 1, wherein, The signal dynamic segmentation and windowing in step 3 specifically comprises: judging whether the signal filtered in step 2 is a stationary signal, if yes, selecting a Hanning window for windowing segmentation; if no, selecting a Blackman window for windowing segmentation.

4. The method for harmonic detection of a power system according to claim 3, wherein, The threshold for judging whether the signal filtered in step 2 is a stationary signal is updated in real time, and the threshold is dynamically corrected according to historical data or environmental changes.

5. The method for harmonic detection for power systems as claimed in claim 1, wherein, A dual-core processor is adopted, and the main CPU and the slave CPU simultaneously process the signal segmented and windowed; the main CPU performs fast Fourier transform calculation, harmonic coarse extraction and spectrum leakage compensation; the slave CPU performs fractional Fourier transform calculation, fractional order optimization, inter-harmonic detection and least square method parameter correction.

6. A method for harmonic detection in power systems as claimed in claim 1 or 5, wherein, In step 4, the main CPU performs fast Fourier transform calculation, harmonic coarse extraction and spectrum leakage compensation on the windowed signal, specifically comprising: the main CPU reads the signal segmented and windowed; the main CPU performs fast Fourier transform calculation on the read signal; the main CPU performs harmonic coarse extraction on the fast Fourier transform calculation result; the main CPU performs spectrum leakage compensation on the coarse extracted harmonics, and the compensation method adopts a dual-spectrum line interpolation method.

7. The method for harmonic detection of power system according to claim 1 or 5, wherein, In step 5, the slave CPU performs fractional Fourier transform calculation, fractional order optimization, inter-harmonic detection and least square method joint optimization of the parameters of the fast Fourier transform and the fractional Fourier transform on the signal segmented and windowed, specifically comprising: the slave CPU reads the signal segmented and windowed; the slave CPU performs fractional Fourier transform calculation on the read signal; the slave CPU performs fractional order optimization on the fractional Fourier transform calculation result, and the optimization target is to maximize the energy concentration degree of the inter-harmonic component; the slave CPU performs inter-harmonic detection on the optimized result.

8. The method for harmonic detection of power system according to claim 1, wherein, In step 6, according to the outputs of the fast Fourier transform and the fractional Fourier transform, least square parameter correction is performed to optimize the parameters of the initial spectral analysis and the inter-harmonic extraction; specifically including: performing least square parameter correction on the outputs of the fast Fourier transform in step 4 and the fractional Fourier transform in step 5, and the correction target is to minimize the residual sum of squares between the fast Fourier transform and the fractional Fourier transform results; performing amplitude optimization on the corrected parameters; performing frequency optimization on the optimized amplitudes; and performing phase optimization on the optimized frequencies.

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