## 2 Literature Overview
The objective function presented here functions as a general framework; however, it is worth noting that it does not fully incorporate additional factors related to the positioning of the APF connection. Acknowledging these factors may enhance the overall understanding of the APF's performance and its effects in different scenarios. The possibility of improving this objective function can be achieved by introducing a set of multipliers. Modified versions of this function consider additional factors, such as preferred buses for APF installation due to accessibility or ease of installation as well as minimizing the maximum APF current [40]. The objective function, shown in expression (2), is related to the voltage or current _THD_. Minimizing the maximum value of _THD_ or _THD_ depends on the reference standards cited by the authors [12; 23; 28; 38; 40; 41; 43; 44; 45; 46; 48]. Different modifications of this objective function involve minimizing these factors at the PCC point, at each bus, or as the total sum of factors across all buses. In addition to the factors mentioned, _THD_ and _THD_, the Total Demand Distortion (_TDD_) factor [41; 42] is also commonly applied in objective functions. Expression (3) refers to the objective function aimed at minimizing power losses, whether they are related to basic losses or harmonic losses [4; 12; 28; 33; 42]. Minimizing higher voltage harmonics at all buses is shown in expression (4), while expressions (5) and (6) refer to reducing financial costs. The expression given in (5) focuses on minimizing the number of APFs and does not have precisely defined values of other vital parameters. Such an objective function requires additional constraints or the introduction of an additional objective function. Some authors assume, in the function, that the expression given in (1) leads to financial cost minimization, assuming that the cost is proportional to the APF rating. A sophisticated approach considers fixed and variable costs as given in expression (6), using coefficients \(c_{\text{fix}}\) as fixed and \(c_{\text{var}}\) as variable costs of the APF. Modifications of the optimization function that consider costs related to investment, as well as operational and maintenance costs, are presented in papers [12; 32]. \[\text{min}f=\sum_{n=1}^{N}|I_{\text{APF},n}|;\;\forall n\in N \tag{1}\]  
\[\text{min}f=\sum_{n}^{N}|THDV_{n}|;\;\text{min}f=\sum_{n}^{N}|THDI_{n}|\;;\; \forall n\in N \tag{2}\]  
\[\text{min}f=\sum_{i=1}^{I}|P_{i}|;\;\forall i\in I \tag{3}\]  
\[\text{min}f=\sum_{w=1}^{W}\sum_{h=2}^{H}|V_{n,h}|;\;\forall n\in N,\;\forall h\in H \tag{4}\]  
\[\min_{x}f=Z_{\text{APF}} \tag{5}\]  
\[\text{min}f=\sum_{n=1}^{N}\left(c_{\text{var}}|I_{\text{APF},n}|-c_{\text{fix}}\right) \tag{6}\]  
For the previous expressions, (1)-(6) stands \(I_{\text{APF},n}\)--defined as the sum of current RMS values of all installed APFs, \(n\)--number of APFs, \(h\)--harmonic order number, \(N\)--maximum number of buses where an APF can be installed, \(M\)--total number of buses in systems (\(M\geq N\)), \(c_{\text{fix}}\) and \(c_{\text{var}}\)--coefficients for fixed and variable costs, and \(Z_{\text{APF}}\)--total number of APFs. By the reviewed literature, the constraints associated with the objective functions can generally be divided into two groups. The first group consists of the physical constraints of the APFs themselves, such as the following:  
* The effective values of the APF currents must be greater than the lower limit (\(|I_{\text{APF}}|min\)) and less than the upper limit (\(|I_{\text{APF}}|max\)) that the filter can provide. * The effective values of the successive current harmonics of the APF are limited according to the physical limits of the APF. * Commercially available sizes of APFs can be considered by introducing the size of the basic unit of the APF.