*Energies* **2020**, *13*, 2776

#### **3. Methodology**

#### *3.1. Formulation of the Design Problem*

The single objective optimisation function is used to find the optimum solution corresponding to the minimum or maximum value defined by the objective function. In contrast, multi-objective optimisation combines two or more individual objective functions to determine a set of trade-off solutions, which allow decision makers to select the most suitable solution based on the problem requirements [60]. In this study, in sizing optimisation methodology depending on the requirements of the power plant designer, each of the two objectives can be used to produce an optimal design for the PV power plant. In addition, for comparison purposes, the optimum values are calculated by using each objective function individually to evaluate the PV power plant performance.

Furthermore, multi-objective optimisation can be used in the design of PV systems with a small capacity in the range of kW, with a small number of PV modules and inverters or in hybrid renewable energy systems for example (PV-wind) or (PV-diesel denerator-battery). However, in large scale PV power plants (i.e., >200 kW nominal power rating—the largest plants reaching several tens of MW of capacity), with a considerable number of components required in PV plant installation, it is well-known that the levelised cost of energy (LCOE) is applied to enable the reduction of the PV plant cost per watt of nominal power that is installed [61,62], for this reason, single objective optimisation is used. Additionally, a recent study is presented in [63] to investigate the LCOE of large scale PV power plants at 8 PV plants ranging from 1 to 46 MWp and many similar studies can be found in the literature.

In this section, two objective functions are considered to evaluate the PV power plant performance and to solve its complex design problem. The design variables and constraints of the proposed methodology are also explained.

#### 3.1.1. Objective Function

In this work, the LCOE and maximum annual energy were set as objective functions to determine the optimal solution of the PV plant design. These two objective functions can be combined to form a single optimisation function.

The first part presents the LCOE which is calculated on the basis of the sum of maintenance, operation and installation costs of the plant divided by the total energy generation of the plant during its lifetime. The LCOE method is generally applied to compare power plants with different energy generation sources, by considering the appropriate cost structures. However, the best LCOE for power plants presents the lowest possible investment with high annual energy production. The second part presents the maximum amount of annual energy that can be captured by the PV modules during the PV plant in its lifetime, which is 25 years. The single optimisation function is expressed by the following equation:

$$\min\_{X} \left[ \left( \frac{\mathbb{C}\_{c}(X) + \mathbb{C}\_{M}(X)}{E\_{\text{tot}}(X)} \right) \cdot a - \left( (1 - a) \cdot \left( P\_{\text{plunt}}(X) \cdot n\_{\text{s}} \cdot EAF \right) \right) \right] \tag{1}$$

where *ns* is equal to 1 year.

The optimum values are calculated by using each objective function individually. In other words, in the objective function, *a* is a binary number; if *a* is equal to 0, the target of the objective function is maximum energy and, if *a* is equal to 1, the objective function target is minimum LCOE.

#### 3.1.2. Design Variables

The proposed optimisation algorithm was used for the calculation of all the decision variables, to determine the optimum design of the PV power plant. The chosen optimisation algorithm should have high performance in determining the best design variables and solving the design problem. In this methodology, the proposed decision variables, including the number of PV modules connected

in series (*Ns*) and parallel (*Np*), number of PV module lines per row (*Nr*), the distance between two adjacent rows (*Fy*), the tilt angle of the PV module (β), the orientation of PV modules (*PVorien*), that can be installed vertically or horizontally, optimum PV module (*PVi*), and inverter (*INi*), can be selected on the basis of several alternatives from a list of possible candidates.

The vector of the decision variables are summarized as given by the following expression:

$$X = \begin{bmatrix} N\_s \ N\_p & N\_r \ \beta \ F\_y P V\_{orien} & P V\_i \ IN\_i \end{bmatrix} \tag{2}$$
