## 5 Discussion and Outlook  
The simulation results shown in Figure 9 clearly demonstrate that stable and high-quality field-null configuration is not only designed but also has the potential to be realized in real experiments. This provides an exciting possibility of using this method on HL-3 to achieve more robust plasma breakdowns despite varying conditions, like pre-gas fueling, wall conditions and loop voltages. Yet, it is also clear that more verifications are required before actually using it in experiments with enough confidence. Firstly, the toroidally axisymmetric 2-D model of the vacuum vessel applied here could differ from the 3-D reality due to the existence of windows, pipes and first wall modules, etc. Thus, more careful calibration of the model parameters needs to be performed using supplementary diagnostic data, like magnetic probes and flux loops, etc. The construction of 3-D models, like using the finite-element method (FEM) [4], will be of great help in raising the accuracy in calculating passive eddy currents but is more complex than the presented model and is left for future work. On the other hand, the models of power supplies can also be improved to include the characteristics of thyristor rectifiers as they are on HL-3 [29], which could help in resolving the existing oscillation problems when performing FB control of CS and PF coils in experiments. Finally, targeted experimental tests are definitely needed to verify the optimized field-null configurations, as well as the applicability of designed coil current waveforms, and to further focus on varying operational scenarios and different boundary conditions to ensure that the configurations are robust and reliable under a wide range of conditions, if sufficient shots are available. Since the \(B_{\perp}\) configuration during the early stage not only affects the breakdown process but also impacts the equilibrium and position control during the subsequent plasma ramp-up stage, it is necessary to consider the optimization problem of \(B_{\perp}\) in a more integrated way, as has been demonstrated in existing references. This could be performed in the proposed method by simply replacing the cost function \(f(\mathbf{I})\) in Equation (8) with other forms. For example, the following \(f(\mathbf{I})\) can be chosen to obtain a vertical field \(B_{Z}\) closed to a wanted value \(B_{Z,0}\) while retaining the field-null characteristic by minimizing \(B_{R}\):  
\[f(\mathbf{I})=\sum_{j}^{j\in\mathbf{A}}\left|B_{R,j}^{2}(\mathbf{I})+ \left(B_{Z,j}(\mathbf{I})-B_{Z,0}\right)^{2}\right|\] \[=\sum_{j}^{j\in\mathbf{A}}\left\{\left[\sum_{i=1}^{105}G_{R,j}l_{ i}\right]^{2}+\left[\sum_{i=1}^{105}G_{Z,ij}l_{i}-B_{Z,0}\right]^{2}\right\} \tag{17}\]  
The optimization problem of \(f(\mathbf{I})\) in Equation (17) is still quadratic and can be solved by transferring it into a linear equation set, similar to the process in Section 3.1. Other constraints, such as the magnitude of coil currents, could also be added in this way or by alternatively introducing boundaries to define the proper parametric domains. This indicates that the approach proposed in this paper is highly expandable and can be improved for various purposes in the future. Real-time algorithms could also be helpful if more signals other than coil currents are involved, such as the data from magnetic probes and flux loops. By developing more complicated real-time algorithms, the control of plasma position or even shape could become possible in the early stages of plasma formation. Unfortunately, the circumstance here is complicated by the quick evolution of early-stage plasmas and large eddy currents in the vacuum vessel, and this part of work is left for the future.