11.3 Weighting Factors Adjustment

The weighting factor tuning procedure will vary depending on which types of terms are present in the cost function, as classified in the previous section.

11.3.1 For Cost Functions with Secondary Terms

This is the easiest case for weighting factor adjustment, since the system can be first controlled using only the primary control objective or term. This can be very simply achieved by neglecting the secondary terms forcing the weighting factor to zero (λ = 0). Hence the first step of the procedure is to convert the cost function with secondary terms into a cost function without weighting factors. This will set the starting point for the measurement of the behavior of the primary variable.

The second step is to establish measurements or figures of merit that will be used to evaluate the performance achieved by the weighting factor. For all the examples given in Table 11.2 a straightforward quantity should be one related to the primary variable, which is current error. Several error measures for current can be defined, such as the root mean square (RMS) value of the error at steady state, or the total harmonic distortion (THD). At least one additional measure is necessary to establish the trade-off with the secondary term. For the three cost functions of Table 11.2 the corresponding measures that can be selected are: the device average switching frequency f_sw, the RMS common-mode voltage, and the steady state input reactive power.

Once the measures are defined, the procedure is as follows. Evaluate the system behavior with simulations starting with λ = 0 and increase the value gradually. Record the corresponding measures for each value of λ. Stop the increments of λ once the measured value for the secondary term has reached the desired value for the specific application, or keep increasing λ until the primary variable is not properly controlled. Then plot the results and select a value of λ that fulfills the system requirements for both variables. This procedure can be programmed by automating and repeating the simulation, introducing an increment in the weighting factor after each simulation.

In order to reduce the n umber of simulations required to find a proper value for the weighting factor, a branch and bound algorithm can be used. For this approach, first select a couple of initial values for the weighting factor λ, usually with different orders of magnitude to cover a very wide range, for example, λ = 0, 0.1, 1, and 10. A qualitative example of this algorithm is illustrated in Figure 11.1. Then simulate these weighting factors and obtain the measures for both terms, M¹ and M², for the primary and secondary terms respectively. Then compare these results to the desired maximum errors admitted by the application and fit them into an interval of two weighting factors ( in the example). Then compute the measures for the λ in half of the new interval (λ = 0.5 in the example) and continue until a suitable λ is achieved. Note in Figure 11.1 that each solid line corresponds to a simulation and dashed lines correspond to values already simulated. This method reduces the number of simulations necessary to obtain a working weighting factor.

Figure 11.1 Branch and bound algorithm to reduce the number of simulations required to obtain suitable weighting factors (Cortes et al., 2009 © IEEE)

The qualitative example of Figure 11.1 can be matched with the results for the common-mode reduction case shown below in Figure 11.3a. Note that with only six simulations the search for λ_cm would have narrowed to an interval where any λ_cm would work properly.

11.3.2 For Cost Functions with Equally Important Terms

For cost functions like those listed in Table 11.3, a different procedure needs to be considered since λ is not allowed to be zero.

As a first step the different nature of the variables has to be considered. For example, when controlling torque and flux in an adjustable speed drive application with a rated torque and flux of 25 N m and 1 Wb respectively, the torque error can have different orders of magnitudes making both variables not equally important in the cost function and affecting the system performance. Thus the first step is to normalize the cost function. Once normalized, all the terms will be equally important and now λ = 1 can be considered as the starting point. Usually a suitable λ is located close to 1. Note that the cost functions in Table 11.3 have already included this normalization (nominal values are denoted by subscript n).

The second step is the same as in the previous procedure, that is, measurements or figures of merit have to be defined in order to evaluate the performance achieved by each weighting factor value.

The last step is to perform the branch and bound algorithm of Figure 11.1 for a couple of starting points. Naturally λ = 1 has to be considered, but λ = 0 is not a possible alternative. When a small interval of weighting factors has been reached, meaning, by small interval, that there are no big differences in the measured values obtained with the upper and lower bounds of the interval, then the weighting factor has been obtained.