# Optimization of DC Energy Storage in Tokamak Poloidal Coils
## 5 Optimization Strategies and Criteria
### Available Optimization Formula for PF Coils  
Considering the SC and the converters as ideal, the most simplified balance consists of transferring all the energy stored in the SC to the load. So, the DC energy, \(E_{\text{DC}}\), must be higher than that storable in the load, \(E_{L}\):  
\[E_{\text{DC}}\gg E_{L} \tag{2}\]  
The energy in the SC can be derived from \(V_{\text{max}}\) by (1). However, it is useful to report that the SC capacitance also depends on voltage, with a mostly linear behavior [49; 50] (though more refined approximations could be proposed [51; 52]):  
\[C(V)=C(V_{\text{ref}})+k_{v}[V-V_{\text{ref}}] \tag{3}\]  
This effect is never reported in manufacturers' datasheets but is commonly observed in experimental characterizations [49; 50]. For this reason, (1) can be used to retain the effect in (3) as a safety margin for design.  
If the load is limited to the coil, its (magnetic) energy is simply due to its auto-inductance, \(L\):  
\[E_{L}=\frac{1}{2}{LI_{\text{max}}}^{2} \tag{4}\]  
Table 1 reports the values of auto-inductances of the DTT PF coil corresponding to diagonal entries of the inductance matrix, \(\underline{M}\).  
However, the actual energy also contains other contributions that can be calculated by integrating the powers in Figure 3, as shown in Figure 6. In this way, a scenario energy variable with time \(e(t)\) is obtained (here and in the following, the lowercase letters denote a time-dependent behavior):  
\[e(t)=\int_{0}^{t}p(\tau)d\tau \tag{5}\]  
Figure 6 neglects the negative energies at the end of the operation, because they derive from the basic idea of discharging all the coils in the same time interval (see Figures 2 and 3), which may give more energy back to some coils with respect to others. This would be corrected when the scenarios are conceived, also considering this aspect.  
An interesting single parameter characterizing the scenario energy consists of its maximum over time:  
\[E_{\mathrm{s}}\stackrel{{\mathrm{def}}}{{=}}\max_{t}\{e(t)\} \tag{6}\]  
The values of \(E_{\mathrm{s}}\) and \(E_{L}\) coincide for a single coil without mutual interactions.  
The definition of \(E_{\mathrm{s}}\) allows the introduction of the new concept of efficiency of the DC storage, that is, the portion of available energy that is actually used for the load:  
\[\eta_{\mathrm{DC}}\stackrel{{\mathrm{def}}}{{=}}\frac{E_{ \mathrm{s}}}{E_{\mathrm{DC}}} \tag{7}\]  
It is important to stress that, while a high \(\eta_{\mathrm{DC}}\) means a full exploitation of the SC bank, on the other hand, a lower value can provide a safety margin that could be useful to compensate for unexpected and not considered effects (for example, the internal losses and drops in the semiconductors).  
\begin{table}
\begin{tabular}{c c c c} \hline
**PF Coil** & \(L\) & \(E_{L}\) & \(E_{\mathrm{s}}\) \\ \hline
**PF1** & 454 mH & 80 MJ & 47 MJ \\ \hline
**PF2** & 298 mH & 65 MJ & 54 MJ \\ \hline
**PF3** & 690 mH & 46 MJ & 53 MJ \\ \hline
**PF4** & 690 mH & 269 MJ & 66 MJ \\ \hline
**PF5** & 298 mH & 77 MJ & 88 MJ \\ \hline
**PF6** & 454 mH & 185 MJ & 100 MJ \\ \hline \end{tabular}
\end{table}
Table 1: Nominal auto-inductances and possible magnetic energy storable in the DTT PF coils. The energy calculated using (4) is compared with the \(E_{\mathrm{s}}\) obtained by the complete MHD model.  
Figure 6: Energies delivered to the PF coils in the DTT reference scenario.  
The values of energies estimated by the different approaches are compared in Table 1. The use of \(E_{\rm DC}\), derived by the complete MHD model, provides a remarkable reduction for the estimation of the energy flowing in the PS circuits, and thus, a potential increase in the achievable efficiency.