# Solution of Orifice Hollow Cathode Plasma Model Equations by Means of Particle Swarm Optimization
## 5 Conclusions  
The paper has investigated the feasibility of using the Particle Swarm Optimization technique to solve a non-linear system of equations describing the plasma physics within orifice hollow cathodes. The results of the assessment test cases, conducted on Sitael AlphCA, Pisa, Italy, have shown that the coupling between PSO and plasma model reveals itself as a fast, powerful and accurate preliminary design and rebuilding tool for OHCs. The trends of all the unknowns of the problem are well predicted. Figure 8: PSO targeting the experimental measurements (_test3_) [26] of the discharge voltage as function of mass flow rate at 2A. Figure 9: Comparison of the expected emitter temperatures when PSO is targeting the experimental measurements [26] of the discharge voltage and when the discharge voltage is free, both as functions of mass flow rate at 2A. Particularly, the discharge voltage as a function of mass flow rate is predicted with a confidence bound in the range \(2\div 5\) V for the current span examined. The computations at 2A and 3A well match the experimental data, whereas at 1A and now mass flow rates, the discharge current is underestimated (maximum by 40%), probably because the system of equation is inadequate to describe the plasma physics in those conditions. The discharge current trend as a function of mass flow rate is of a parabolic type for all currents: it initially decreases, as expected, but it increases at high mass flow rates. This could be due to the absence of a double sheath at the cathode orifice exit in the model, which might dampen the discharge voltage increase at high mass flow rates. All the other unknowns of the problem are computed with a very tiny confidence bound with the exception of the plasma density in the emitter region (\(\pm 20\%\) of the average value) and the wall temperature of the emitter and the orifice (\(\pm 50\) K respect to the average values). Regarding plasma densities, it is worth remarking that the PSO has been able to get within a range of three orders of magnitude of the right one for neutral and plasma densities. It has been verified that the confidence bound of the emitter temperature can be reduced and its trend can be improved by targeting the discharge voltage. The difficulty in finding solutions to the complex system of equations characterizing the plasma within the OHC has been fully overcome with the usage of the PSO technique. This enables the possibility of easily performing sensitivity studies on all the geometrical parameters of the cathode model, as well as quickly testing model modifications (i.e., the addition of the aforementioned double sheath at the cathode orifice exit). As future development, a thermal model in FEMM [27] will be coupled to the PSO-plasma model, according to ref. [11]. This would give the wall temperatures a new constraint; thus, more accurate plasma properties predictions are expected. Conceptualization M.P. ; methodology, G.C. and M.P. ; software, G.C. ; validation, G.C. ; formal analysis, G.C. ; data curation, M.P. and G.C. ; writing--original draft preparation, G.C. nd M.P. ; writing--review and editing, F.B. ; visualization, G.C. ; supervision, F.B. and M.P. ; project administration, F.B. All authors have read and agreed to the published version of the manuscript. This research was funded by the Italian research program in aerospace, PRORA DM662/20 (Programma Operativo Ricerche Aerospaziali) entrusted by MIUR (Ministero dell'Istruzione, Ministero dell'Universita e della Ricerca). Not applicable. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy. The authors wouldlike to thank [PERSON], [PERSON] and [PERSON] for the internal review of the paper. The authors declare no conflicts of interest. The following abbreviations are used in this manuscript:  
OHCs