## 4 Application to Power Electronics
### Specifications in Terms of the Optimization Problem
#### 4.2.2 Optimization Constraints  
The aim of the optimization endeavor is to minimize the mass of the aircraft's electrical power channel. This intricate optimization problem is governed by several constraints, as follows:  
* Geometric constraint: the Routh Criterion, which must remain positive. To compute this value, a global equivalent circuit of the aircraft power channel was established, based on an RL series circuit [11]. It was achieved by bringing all the resistance and inductance components to the continuous side. Therefore, the Routh Criterion is applied in this linear continuous system, which represents the global behavior of the system;
* Frequency-domain constraints: The input current (\(iL2\)) of the channel must adhere to the standards delineated in Table 2, corresponding to low frequencies (up to 14,800 Hz). IHR signifies the Individual Harmonic Rate, as articulated in Equation (5) [48];
* Time-domain constraint: Upon achieving a steady state, the 'Load' resistance depicted in Figure 11 is diminished by one-eighth of its size to induce a disturbance and initiate a new transient state. During this transient phase, the channel's output voltage (\(Vdc\)) must conform to the template illustrated in Figure 12. At each computational interval, should the output voltage (\(Vdc\)) deviate from the prescribed template, the constraint value is augmented by the discrepancy between the actual output and the template's bounds. For the constraint to be satisfied, it must equate to zero. The gradients of this constraint concerning the optimization inputs are derived through automatic differentiation, as no analytical formula exists. \[\text{IHR}_{\text{k}}\left(\widetilde{\text{x}}\right)=\left|\frac{\widetilde {\text{x}}_{\text{k}}}{\widetilde{\text{x}}_{1}}\right|\] (5)  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Harmonic Order (k)** & **Upper Bound (in \%)** \\ \hline k = 3 & \(\text{IHR}_{\text{k}}<2\) \\ k = 11 & \(\text{IHR}_{\text{k}}<10\) \\ k = 5, 13 & \(\text{IHR}_{\text{k}}<8\) \\ k = 17, 19 & \(\text{IHR}_{\text{k}}<4\) \\ k = 7, 23, 25 & \(\text{IHR}_{\text{k}}<3\) \\ K = 29, 31, 35, 37 & \(\text{IHR}_{\text{k}}<30/\text{k}\) \\ K = 2, 4 & \(\text{IHR}_{\text{k}}<1/\text{k}\) \\ k is a multiple of 3 or is odd & \(\text{IHR}_{\text{k}}<2\) \\ Other cases & \(\text{IHR}_{\text{k}}<2\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Frequency-domain constraints for IHR (iL2).  
Figure 12: Template for the time-domain constraint.  
Here:  
* \(\widetilde{x}1\) is the FFT of x when the frequency is equal to the fundamental one (400 Hz);
* \(k\) is the harmonic order (from \(k=2\) for \(f=800\) Hz, to \(k=37\) for \(f=14800\) Hz).  
The optimization problem encompasses 10 inputs, 1 objective function to minimize, and 38 constraints, which include the following:  
* 1 equality constraint (the time-domain constraint);
* 36 inequality constraints for the frequency-domain constraints (each corresponding performance constrained by an upper bound);
* 1 inequality constraint for the Routh Criteria, which is only constrained by a minimum value (it must be positive).  
In Table 2:  
* IHR (\(\widetilde{i}\widetilde{L}\)) means the Individual Harmonic Rate of the channel input current \(iL2\).  
It is noteworthy that the final value of the output channel voltage varies with each simulation, contingent upon the values of the optimization inputs.