4.1.2. Linear Programming

Linear programming (LP) is a method that uses a mathematical model with linear mathematical relationships for optimizing the objective function(s). LP is used in power system optimization problems to obtain optimal sizes of DG units, because it provides precise solutions [2,56,57]. In Altintas et al. [63], the authors proposed a two-objective LP algorithm to incorporate solar and wind renewable DGs as well as BESS into distribution system expansion planning. The objective minimises the total cost of investment and carbon emissions. This algorithm performed a sensitivity analysis test on the effect of investment costs with respect to wind and solar DGs and BESS. Alturki et al. [64] presented an LP method to obtain optimal hosting capacity of a distribution grid with the objective to maximise the PVDG power using some fundamental variables and to minimise total cost using some uncertain criteria. The results revealed that the computation time for the proposed LP algorithm was very small, especially for large-scale problems. However, the network harmonic level and stability were not considered for evaluation in these works.

#### 4.1.3. Mixed-Integer Linear Programming

The mixed-integer linear programming (MILP) method uses a mathematical model with linear objective function and linear constraints in which, at the minimum, one design variable must be an integer. The implementation of MILP is difficult in large-scale problems because it uses too much computation time. In Santos et al. [1], MILP is applied to determine the optimal locations, sizes and timing of smart-grid technologies for minimising the net present value of the total cost and for maximising the renewable DG integration. In Mishra et al. [65], a chance-constrained stochastic MILP algorithm is modelled to determine optimal investment decisions of DGs considering operational uncertainties, while an evolutionary vertical sequencing protocol algorithm is used to further optimize the objective function that minimises the total cost of investment and operation. Santos et al. [66] proposed an improved model aimed at optimizing the system operation in a coordinated way, where distributed renewable energy sources (DRES), energy storage systems (ESS) and distribution network system reconfiguration (DNSR) are considered along with the uncertainty of the resources. The objective function was modelled to incentivize the uptake of DRES by considering the cost of emissions to decarbonize the power system. In Ajeigbe et al. [67,68], the authors applied the MILP algorithm to maximise the optimal allocation of solar, wind and biomass DGs into the distribution system by minimising the NPV of total cost and by confining the small signal stability of the networks to a required level. All the works reviewed here modelled uncertainties of renewable energy resources and evaluated voltage stability of the network but were not able to evaluate the impact of BESS/RERDG powers on the harmonic contents of the networks. Likewise, their results did not report global optimal solutions to BESS/PVDG optimal allocation problems.

LP and MILP suffer from a lack of flexibility. They normally require pre-conditions such as convexity, linearity and continuity of objective functions, which are difficult to meet in practice [2,57].

#### 4.1.4. Nonlinear Programming

Nonlinear programming (NLP) is a mathematical programming method that uses nonlinear objective function and solely continuous variables and constraints. The NLP computation involves the differentials of objective functions and constraints. In solving nonlinear problems, a search path is selected iteratively by defining the starting partial differentials of the problem equation. This approach could be based on first-order or higherorder methods such as the reduced gradient method [69,70] and other search methods [71,72], Newton Raphson method [73] and successive quadratic programming [74,75] which are used for solving DG allocation planning problems.

#### 4.1.5. Mixed-Integer Nonlinear Programming

Mixed-integer nonlinear programming (MINLP) utilizes a mathematical model with nonlinear objective functions and constraints and both continuous and discrete variables. MINLP algorithms have been applied in power systems to determine the optimal sizes and locations of DGs and BESSs. Some of the disadvantages of MINLP are long computation time and a very large number of decision variables [2,56,57]. Salyani et al. [76] applied MINLP in the mathematical modelling for the simultaneous optimal allocation planning of high- and medium-voltage substations, robust medium-voltage feeder routing and renewable DG units. The authors used adaptive GA to find optimal locations and sizes while the uncertainties of renewable DGs, fuel prices, electricity and demand were evaluated. A mixed-integer nonlinear programming-model-based methodology is presented in Valencia et al. [11] for the optimal location, selection, and operation of BESSs and renewable distributed generators (DGs) in medium–low-voltage distribution systems.

#### 4.1.6. Fuzzy Logic

The fuzzy logic (FL) method was developed in 1979 to solve power system problems. The FL method is based on the concept of a classical set, such as the identification of a membership function that is associated with each member as indicated by a binary number 0 and 1 [77]. The membership function dictates the resemblance level of a member in a fuzzy subset. Some of the common membership functions are the triangular, trapezoidal, piecewise-linear and Gaussian functions [2,57,59]. In Injeti and Kumar [78], FL is applied to DG allocation problems, with minimisation of power losses and improvement in voltage profiles as the objective function. Sharma et al. [79] proposed a FL controller in determining the optimal sizes and locations of DGs in order to minimise power losses and to enhance loadability and voltage profiles of distribution networks. However, the results from these works did not report the optimality of their solutions, the evaluation of network stability or harmonic contents.

The works discussed thus far on FL have not considered the impact of DGs and BESS on the oscillatory modes and harmonic contents of the distribution networks. To achieve practical solutions, dynamic networks must be simulated for the evaluation of distribution system stability and harmonic contents.

#### *4.2. Intelligence Search Methods for BESS/PV Distributed Generations*

Artificial intelligence (AI) is the application of human intelligence to perform tasks in machines [59]. AI is applied in the intelligence search methods (ISM) used in power systems for optimal sizing and placement of DGs. Intelligence search methods are heuristics algorithms that fasten up the processes of obtaining near-optimal solutions for complex and large DG problems. The advantages of intelligence search methods over other conventional methods is the simplicity of implementation and robustness. However, the accuracy and precision of ISMs are not reliable. They usually take much computation effort [2,56,57,80]. Some of intelligence search methods are presented below.
