1(globalThis.webpackChunk=globalThis.webpackChunk||[]).push([[1818],{21818(e){e.exports="<h1>Summary</h1>\n<p>The classic method of accurately finding the max-power point by fitting a parabola to the power-versus-voltage curve in the vicinity of the max-power point is not the best approach. With the same computational simplicity, a parabolic fit of power-versus-conductance provides a more accurate estimate of both V<sub>max</sub> and P<sub>max</sub>. When used in the context of numerical device simulation, the conductance method reduces the number of bias points that need to be calculated, and this method is used in PC3D. The equations that are needed to fit a parabola through three data points surrounding P<sub>max</sub> are given in the appendix below.</p>\n<p>When the solar cell is known to be well-represented by a two-diode circuit model, even greater accuracy can be obtained by solving for the circuit parameters that best fit the entire I-V curve. However, this approach adds significant numerical complexity, and convergence of the parameters is not guaranteed. Furthermore, this approach should be avoided when there is a possibility that the solar cell is poorly represented by the circuit model, as is often the case when dealing with experimental data.</p>\n<h1>Appendix: Parabolic Fit</h1>\n<p>To find the parabola y = Ax<sup>2</sup> + Bx + C that passes through three data points (x<sub>1</sub>,y<sub>1</sub>), (x<sub>2</sub>,y<sub>2</sub>), and (x<sub>3</sub>,y<sub>3</sub>), where x<sub>1</sub> â x<sub>2</sub> â x<sub>3</sub>, first calculate a denominator term D, and use that to find the coefficients A, B, and C:</p>\n<p>$$<br>\n\\begin{equation}<br>\nD = (x_1-x_2)(x_2-x_3)(x_1-x_3)<br>\n\\end{equation}<br>\n$$</p>\n<p>$$<br>\n\\begin{equation}<br>\nA = [(x_2-x_3)(y_1-y_2)-(x_1-x_2)(y_2-y_3)]/D<br>\n\\end{equation}<br>\n$$</p>\n<p>$$<br>\n\\begin{equation}<br>\nB = \\frac{(y_1 - y_2) - A(x_{1}^{2} - x_{2}^2)}{x_1 - x_2}<br>\n\\end{equation}<br>\n$$</p>\n<p>$$<br>\n\\begin{equation}<br>\nC = y_1 - B \\space x_1 - A \\space x_1^2<br>\n\\end{equation}<br>\n$$</p>\n<p>Fitting through three data points surrounding the max-power point is appropriate for numerical simulations. For experimental data, it is better to fit the parabola through many data points to average-out the noise. To minimize the mean-squared-error, start by calculating some sums over the N data points (x<sub>i</sub>,y<sub>i</sub>) to be included in the fit:</p>\n<p>$$<br>\n\\begin{equation}<br>\nX_0 = N, \\space X_1 = \\sum x_i, \\space X_2 = \\sum x_i^2, \\space X_3 = \\sum x_i^3, \\space X_4 = \\sum x_i^4,<br>\n\\end{equation}<br>\n$$</p>\n<p>and</p>\n<p>$$<br>\n\\begin{equation}<br>\n\\space Y_0 = \\sum y_i, \\space Y_1 = \\sum y_i x_i, \\space Y_2 = \\sum y_i x_i^2<br>\n\\end{equation}<br>\n$$</p>\n<p>Calculate a denominator term D, and use that to find the coefficients A, B, and C:</p>\n<p>$$<br>\n\\begin{equation}<br>\nD = X_4(X_2X_0-X_1X_1) - X_3(X_3X_0-X_2X_1) + X_2(X_3X_1-X_2X_2)<br>\n\\end{equation}<br>\n$$</p>\n<p>$$<br>\n\\begin{equation}<br>\nA = [Y_2(X_2X_0-X_1X_1) + Y_1(X_1X_2-X_3X_0) + Y_0(X_3X_1-X_2X_2)] / D<br>\n\\end{equation}<br>\n$$</p>\n<p>$$<br>\n\\begin{equation}<br>\nB = [X_0(Y_1-AX_3) - X_1(Y_0-AX_2)] / (X_2X_0 - X_1X_1)<br>\n\\end{equation}<br>\n$$</p>\n<p>$$<br>\n\\begin{equation}<br>\nC = (Y_0 - AX_2 - BX_1)/X_0<br>\n\\end{equation}<br>\n$$</p>\n<p>Whether for three data points or N data points, if A<0 the parabola has a maximum, and the peak of the parabola is located at x<sub>max</sub> = -B/(2A) with value y<sub>max</sub> = C - B<sup>2</sup>/(4A). When using the conductance method, the x values are the I/V ratio at each point on the I-V curve and the y values are the power at each point, so that P<sub>max</sub> = y<sub>max</sub> and (V<sub>max</sub>)<sup>2</sup> = y<sub>max</sub>/x<sub>max</sub>. If the peak is found to be located outside the range of the x values (extrapolation), then try to select a more appropriate range of data points to encompass the max-power point.</p>\n"}}]);
Line numbers count LF bytes from the start of the resource, as the search results do. Vendor segments are library code the classifier recognised; they are stored but not indexed. Bytes are shown as Latin1 characters, one per byte.