*3.1. Model Calibration*

As can be found from Section 2, there are several model parameters to be calibrated. Engine geometrical parameters are extracted from the engine project guide. For the model parameters available in the turbocharger model, they are calibrated by using the performance map. Model parameters in several engine component sub-models, such as air cooler cooling e fficiency and pressure drop, are estimated by using the engine shop trial report, which provides the measured engine performance parameters at several steady loading conditions. The last four model parameters remained to be determined include *kCA*0, *kCA*1, *k*ζ0 and *k*ζ1, which have significant influences on the MVEM's predictive accuracy. For estimating the four model parameters based on the measured data provided in engine shop trial report, the Parameter Estimation toolbox provided by MATLAB/Simulink is used, which converts the model parameter estimation problem to numerical optimization problem. In general, the model parameter calibration procedure is carried out in three steps as following:

(1) Initialization of *kCA*0 and *kCA*1. In this step, it is assumed that . *msz* is equal to the compressor mass flow rate . *mc*, which, in turn, can be estimated by using the compressor model. Based on this assumption, the initial value of *CzAz* at each engine loading condition can be estimated by using Equation (1), and then the initial value of *kCA*0 and *kCA*1 can be estimated;

(2) Initialization of *k*ζ0 and *k*ζ1. For estimating the initial value of ζ at each engine loading condition, the Parameter Estimation toolbox is adopted for the whole engine model. Non-linear least square method is selected as the optimization method. The initial value of *CzAz* obtained in step 1 is regarded as known quantity in this step. The input variable required only includes the engine speed, whereas the output variables include pressure and temperature in the scavenging and exhaust manifolds as well as the turbocharger rotational speed. The sum of squares of the errors between the predicted and measured value of the selected output variables is used as the cost function. As a result, the initial value of ζ at each engine loading condition can be estimated. Consequently, *k*ζ0 and *k*ζ1 can be initialized;

(3) Final calibration of *kCA*0, *kCA*1, *k*ζ0 and *k*ζ1. The four model parameters are calibrated simultaneously in this step, the values of which obtained in step 2 are used as the initial values. Note that the calibration process in this step is carried out by using the measured data at all the engine loading conditions simultaneously with the cost function as shown in Equation (26).

$$V(\varphi) = \frac{1}{NS} \sum\_{i=1}^{S} \sum\_{n=1}^{N} \left( e^i[n] \right)^2 \tag{26}$$

where *N* is the number of measured loading points available in the engine shop trial report; *S* is number of selected output variables; ϕ represents the parameters to be estimated.
