1. 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.

2. 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!

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

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

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

6. 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.

7. 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.

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

9. Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Example on a n-dim. regression Experimental or tabulated data: Guessed function: 22 6 2 5 22 4 2 321 ),,(*xzazyxazyayxazyxayazyxf 

10. Analysis, Modelling and Simulation of Energy Systems, SEE-T9 Mads Pagh Nielsen Example (continued) a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 123456 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+03 -30,341.419,002,10E+06 -210.555,00209.122,001,76E+11 -986,381.439,005,88E+06 -267,901.822,004,37E+06 -932,675.201,003,76E+07 -7.390,0229.766,001,38E+09 Squared error sum1,78E+11

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

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

13. 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.

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