Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Modelling of part load conditions (I) Component characteristics and determination.

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Presentation transcript:

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Modelling of part load conditions (I) Component characteristics and determination of such (for pumps compressors, fans etc.). Usage of characteristics in modelling. - Matematical regressions in one- or multiple dimensions. - Interpolation in one and more dimensions (bilinear and higher order methods.

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Determination of characteristics Experimential characteristics Calculated characteristics NOTE: Always investigate the assumptions done making the characteristics! Be 100% sure about what the characteristic shows regarding units and properties!

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen A general fan characteristic

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen General pump characteristic

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen General compressor characteristic

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Usage of characteristics in models Multidimensional regressions describing the characteristic. Setting up multiple non-linear equations for different intervals. Use tables and interpolate-/extrapolate non tabulated points.

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Multidimensional regressions The method of least squares Strategy: Minmize the sum of the quadratic error as function of independent parameters of a guessed expression. That means: The method can be formulated as an optimization problem.

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Regression of n-dim. function with m unknowns Meassured values Guessed function

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Example on a n-dim. regression Experimental or tabulated data: Guessed function: ),,(*xzazyxazyayxazyxayazyxf 

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Example (continued) a1a1 a2a2 a3a3 a4a4 a5a5 a6a We give in guessed for the coefficients: The result is evaluated: f(x,y,z)f*(x,y,z) estimate(f(x,y,z)-f*(x,y,z))² 40,85126,007,25E , ,002,10E , ,001,76E , ,005,88E , ,004,37E , ,003,76E , ,001,38E+09 Squared error sum1,78E+11

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Flow-chart for regression-procedure Flow diagram for multi-dimension R²-regression: Is error minimized? Determine squared error sum R² Guess coefficients Choose function with N-coefficients Yes Is the error still too big to use the function? Then try a new expression! Complex functions typically have to be modelled in intervals! Optimization algorithm

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Bilinear interpolation

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Interpolation in multi-dimension: Strategy: The fundamental idea is to split up the process into several subsequent 1D interpolations. We wish to find P(x 1,x 2,...x n ): Determine the surrounding points (search the ordered table to determine the location in the table). Make a 1D interpolation in (n-1) directions to determine the interior points in the ”row-plane”. Make at last an interpolation in the ”column-plane” inbetween the found points in the ”row-plane” to find the function value P.

Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Interpolation in n dimensions: Yes