Abstract
The analysis of harmonic distortions and their propagation in electric power systems is crucial for power quality assessment. Harmonic power flow (HPF) is a widely used technique for this purpose. However, most existing research on HPF focuses on deterministic algorithms, and only a few studies have considered uncertainties associated with harmonic sources connected to microgrids. Due to the lack of recent approaches for evaluating the impact of uncertain data on HPF results, this paper proposes a new methodology for calculating harmonic voltages and distortions in microgrids by assuming uncertain parameters for both loads and harmonic sources. To obtain interval results for harmonic voltages, the load flow equations are expanded up to the second-order terms using Taylor series, considering all partial derivatives of the system variables with respect to a percentage uncertainty value. The proposed method is evaluated using simulations on a 33-bus distribution system, considering the microgrid in both islanded mode and connected to a main power substation. As demonstrated, the interval results are similar to those obtained from Monte Carlo simulations, proving the efficiency of the proposed method with robust solution and reduced computational time.
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de Sousa, L.L.S, de Melo, I.D and Variz. A. M. wrote the main manuscript text and prepared all the figures. All authors reviewed the manuscript.
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A First and second derivatives
A First and second derivatives
The first derivatives of the active and reactive powers (\(P_k\) and \(Q_k\)) in relation to the state variables that make up the Jacobian matrix are presented from Eqs. (35) to (46) where \( G_{km}\) is the conductance of the line connecting the bus k to m; \(B_{km}\) is the susceptibility of the line connecting the bus k to m; \(V_k\) is the magnitude of the voltage on bus k and \(\theta _k\) are the stress angles on bus k.
The active power generation at PV buses is modelled conform in which Eq. (1) which considers the steady state droop frequency regulation mathematical for all generating units. Thus, the partial derivative of the frequency control equation in relation to the active power generated by the DG and in with respect to the system frequency corresponds to (47) and (48), respectively.
As \(y'_k\), is not a function of \(\theta _k\), \(V_k\) and \(Q_g\) then the corresponding partial derivatives are given by:
The partial voltage derivatives of each DG in the system correspond to:
The slack bus is a single bus in the network and can be chosen for any bar i, where the partial derivatives with respect to \(\theta _i\), \(V_k\), \(P_{g_k}\), \(Q_{g_k}\) and f are given by:
The residues are formed according to Eqs. (54) to (58).
Equation (2) is then solved and the state variables are updated at each iteration it, as seen from Eqs. (59) - (63), as follows:
The second derivatives of the active and reactive powers with respect to the state variables are provided by Equations from (64) to (117):
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de Sousa, L.L.S., Melo, I.D. & Variz, A.M. Interval harmonic power flow for microgrids: an approach using the second-order Taylor series. Electr Eng (2024). https://doi.org/10.1007/s00202-024-02341-8
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DOI: https://doi.org/10.1007/s00202-024-02341-8