A Linearization of DFT Spectrum for Precision Power Measurement in Presence of Interharmonics

The presence of interharmonics in power systems can lead to asynchronous sampling, a phenomenon further aggravated by shifts in the fundamental frequency, which significantly degrades the accuracy of power measurements. Under such asynchronous conditions, interharmonics lose orthogonality with the fundamental and harmonic components, giving rise to additional power components. To address these challenges, this paper introduces a linearization algorithm based on DFT spectrum analysis for precise power measurement in systems containing interharmonics. The proposed approach constructs a system of linear equations from the DFT spectrum and solves it through efficient matrix operations, enabling accurate extraction of interharmonic components near the fundamental and harmonic frequencies (with a frequency interval $\geq$1 Hz). This allows for precise measurement of power across the fundamental, harmonic, interharmonic, and cross-power bands, as well as total power. Test results demonstrate that the proposed method accurately computes various power components under diverse conditions–including varying interharmonic/fundamental/harmonic intervals, fundamental frequency deviations, and noise. Compared to existing methods such as fast Fourier transform (FFT), Windowed interpolation FFT, and Matrix pencil-Singular value decomposition, the proposed technique reduces estimation error by several times to multiple folds and exhibits improved robustness, while maintaining a computational time of only 7 ms for processing 10-power-line-cycle (200 ms) data.

💡 Research Summary

The paper tackles a critical problem in modern power‑quality metering: the presence of interharmonic components, which are non‑integer multiples of the fundamental frequency, causes asynchronous sampling and destroys the orthogonality between voltage and current waveforms. When the observation window does not contain an integer number of cycles of an interharmonic, the classic DFT‑based power calculation (which assumes sinc‑type leakage terms vanish) yields significant errors, especially in the cross‑power terms between fundamental, harmonic and interharmonic frequencies.

To overcome this, the authors develop a linearization of the DFT spectrum. Starting from a multi‑tone signal
(s(t)=\sum_{l=1}^{L} A_l e^{j(2\pi f_l t+\phi_l)})
they show that, for sufficiently large sampling frequency (f_s) and number of points (N), each DFT bin can be expressed as a linear combination of a few nearby bins:
(S_l(k)=\alpha_l \beta_l^{-k}) with (\beta_l = f_l/\Delta f - k) and (\alpha_l = A_l e^{j\phi_l}\frac{e^{j2\pi\beta_l}-1}{j2\pi}).
Because leakage energy concentrates on a small set of adjacent bins, the authors select the bins surrounding each frequency of interest (within at least a 1 Hz interval) and build a system of linear equations (\mathbf{S}= \mathbf{A}\mathbf{x}) where the unknown vector (\mathbf{x}) contains the complex amplitudes and frequency‑position parameters of the underlying tones. Solving this small‑dimensional linear system by direct matrix methods (QR, LU, or least‑squares) yields accurate estimates of the amplitudes, phases, and exact frequencies of the fundamental, harmonic, and interharmonic components.

With these parameters, the power contributed by any pair of voltage‑current components can be computed from the exact analytical expression:
(P_{ab}=U_a I_b \cos