# The Optimization Algorithm of the Forced Current Catholic Protection Base on Simulated Annealing +
## 2 Mathematical Modeling of Auxiliary Anode Optimization Design  
The auxiliary anode is an important part of the cathodic protection system and the grounding grid cathodic protection is more complicated. The model diagram is shown in Figure 1. The potential distribution of the protected body is uniform and the design parameters of the anode are very important. At present, in practical applications, the location of the embedding is determined mainly by experience or field test. In the already operating cathodic protection system, due to improper selection of the position, a protection body at a distance is not protected, and a protection body in the closer vicinity is over protected. The result is relocation and burying of the auxiliary anode, which is neither scientific nor economical. Through theoretical and experimental research on the design parameters of the auxiliary anode, it is found that reasonable anode design parameters should meet potential distribution requirements and ensure a small current density. According to the purpose of cathodic protection design, all protected bodies reach the protection potential. For the protection of metal parts, the surface potential \(\phi\) of the protected metal should meet the purpose of corrosion protection  
\[\phi\leq\phi_{p} \tag{1}\]  
\(\phi_{p}\) is the protection potential, and the value of \(\phi_{p}\) depends on the characteristics of the metal structure to be protected and the medium is selected with reference to relevant experiments and parameters. In order to make the surface potential of all protected bodies in the region meet the above requirements, it can be solved in two ways. One is to rationally arrange and adjust the anode position; the other is to adjust the anode current output. The adjustment of the anode current output is easy to  
Figure 1: Grounding grid cathodic protection model. implement in engineering, as long as the current output of the current source is adjusted. The burying of the anode on the project requires a large investment and engineering quantity, such as the excavation of earthwork and laying of carbon beds. Therefore, it is almost impossible to adjust the position once it is fixed. Therefore, the research goal is to determine the optimal anode embedding position during the pre-construction, anti-corrosion design. The finite element method can usually be used to solve the model. On the selected grid, the potential value at each junction of the cathode surface can be obtained. Let the grounding net have \(n\) nodes and the vector of the potential \(\phi_{j}\) at each node is \(\left[\phi_{1},\phi_{2},\cdots,\phi_{n}\right]^{T}\), marked as \(\overline{\phi}=\frac{1}{n}\sum\limits_{j=1}^{n}\phi_{j}\), and it is defined as  
\[\Psi_{1}[\phi_{e},D_{e}(x_{e},y_{e},h_{e}),L(a,b),n_{e}]=\frac{1}{n}\sum\limits _{j}^{n}\left(\phi_{j}-\overline{\phi}\right)^{2} \tag{2}\]  
\[\Psi_{2}[\phi_{e},D_{e}(x_{e},y_{e},h_{e}),L(a,b),n_{e}]=\left(\overline{\phi }-\phi_{p}\right)^{2} \tag{3}\]  
Here, \(\phi_{e}\) is anode potential, \(D_{e}(x_{e},y_{e},h_{e})\) is the anode position, \(x_{e}\), \(y_{e}\), \(h_{e}\) are the projections of the anode on the X, Y and Z axes respectively,\(L(a,b)\) is the grounding grid specification to be protected and a, b respectively represent the length and width specifications of the grounding grid; \(n_{e}\) is the number of anodes, \(\phi_{p}\) is the best protection potential. In the two functions defined above, \(\Psi_{1}\) represents the degree of uniformity of the potential distribution, and \(\Psi_{2}\) represents the closeness of the calculated value of the average potential to the theoretical optimal value. The following anode optimization problem is proposed: Find \([\phi_{e}^{*},D_{e}^{*}(x_{e},y_{e},h_{e}),L_{e}^{*}(a,b),n_{e}^{*}]\) to satisfy \(\Psi[\phi_{e}^{*},D_{e}^{*}(x_{e},y_{e},h_{e}),L_{e}^{*}(a,b),n_{e}^{*}]=\min \Psi(\Psi_{1},\Psi_{2})\). This represents the optimization of anode position, size and quantity as a multi-objective planning problem.