**4. Objective Function Formulation**

The existing 22 KV Vagarai feeder consists of one standalone 1.5 MW wind turbine (*W*1), and one 1.5 MW wind turbine (*W*2-*PV*) combined in a hybrid configuration with a 0.300 MW solar PV system and one standalone 2.8 MW wind turbine (*W*3). The cost of the entire system and the initial erection and commission cost is fixed. The objective function, which has to be minimized, can be expressed as follows:

$$\text{Min}\,(AC) = N\_{PV}C\_{PV} + N\_{WT}C\_{WT} + P\_{ESS}C\_{ESS} - C\_{\mathcal{S}^5} + C\_{\mathcal{S}^p} \tag{1}$$

where, *AC* is the annualized cost of the system. The annualized total costs of *PV*, *WT* and *ESS* are represented as *CPV*,*CWT* and *CESS*. The total price (\$/year) of electricity purchased from the grid annually is denoted as *Cgp* and the total price of electricity sold to the grid (\$/year) annually is *Cgs*. The annualized cost of each component comprises several parts, that is, annualized capital cost, and annualized operational and annualized replacement cost.

The entire hybrid system comprises several components and the cost of the replacement of components during normal wear and tear and major breakdown is the running cost for operations and maintenance. The salary ofthe operational crew and miscellaneous costs like taxes on land and maintenance of the control room building is also included in the running cost. It also includes the cost ofpurchasing power for the operation of the wind turbines, when synchronized withthe grid. The system with the lowest cost would be classified as possessing the optimal strategy; it should overcome the constraints and one should measure the 'per unit' cost by dividing the total cost and the generation units in KWHr. In the proposed study, the production of power is the total supply of power to the grid at the instance of generation and the stored energyfed back, generated during predictable interruption periods. Hence when measuring per unit cost by dividing the total cost and the generation units in KWHr, it will be less when compared to the cost of the system without energy storage:

$$\text{Max}(Gen) = E\_{W1} + E\_{W2-PV} + E\_{W3} - E\_{\text{Import-V1}} - E\_{\text{Import-V2-PV}} - E\_{\text{Import-V3}} \tag{2}$$

$$\text{Min}(\text{Cost}) = \mathbb{C}\_{W1} + \mathbb{C}\_{W2-PV} + \mathbb{C}\_{W3} - \mathbb{C}\_{\text{Import}-\mathbb{W1}} - \mathbb{C}\_{\text{Import}-\mathbb{W2}-PV} - \mathbb{C}\_{\text{Import}-\mathbb{W3}} \tag{3}$$

Here, the export of energy from the wind turbine (*W*1) is denoted as *EW*<sup>1</sup>*,* the export of energy from wind turbine (*W*2) plus solar Photo Voltaic (*PV*) is denoted as (*EW*2-*PV*) and the export of energy from the wind turbine (*W*3) is denoted as *EW*3. *CW*1 is the cost of the sale of power to grid due to the standalone 1.5 MW wind turbine. *CW*2*PV* is the cost of the sale of power to the grid due to the hybrid 1.5 MW wind turbine combined with the 0.300 MW solar PV systems and *CW*3 is the cost of the sale of power to the grid due to the standalone 2.8 MW wind turbine.

By substituting the real data of the energy supplied to the grid through the 22 KV hybrid feeder of the 110 KV Vagarai SS, into the expressions of (2) and (3), will become:

$$\text{Max(Gen)} = 3016380 + 3362200 + 6570120 - 10800 - 12420 - 11960 = 12913520 \,\text{KWh} \tag{4}$$

$$\text{Min}(\text{Cost}) = 144786.24 + 161385.60 + 354786.48 - 518.40 - 596.16 - 645.84 = 659198 \text{ \\$} \tag{5}$$

In the existing grid-connected hybrid system, without any energy storage, the cost per unit sold without any storage for the year 2018 in the 22 KV hybrid feeder, considering all interruption periods is:

$$C\_{\text{actual}} = \text{Min}\left(\text{Cost}\right) / \text{Max}\left(\text{Gen}\right) \tag{6}$$

By substituting the real data of the net energy sold and the net cost towards supply of energy in (6), one gets the following expression:

$$C\_{\text{actual}} = 659198/12913520 = 0.05104 = 0.051 \,\text{\AA} \tag{7}$$

Hence, the annual average net cost of the energy sold to grid during the year 2018 without any storage system during the interruptions is worked out to be \$0.051.

In the first operational segment, the entire load of the 110 KV Vagarai SS is fully served by the connected hybrid feeder, which consists of three WTz and a 0.300 KW solarPV unit:

$$P\_{W1}(t) + P\_{W2-PV}(t) + P\_{W3}(t) = P\_{Load} \tag{8}$$

In the second operational segment, the entire load is fully served by the connected hybrid feeder, which consists of three WTsz and the 0.300 KW solar PV system, when they are equal and higher than the load demand:

$$P\_{W1}(t) + P\_{W2-PV}(t) + P\_{W3}(t) > P\_{\text{Load}} \text{ or } P\_{W1}(t) + P\_{W2-PV}(t) + P\_{W3}(t) \ge P\_{\text{Load}}\tag{9}$$

The remaining power would be sent to the grid.

In the third operational segment, the load can be partly served by the connected hybrid feeder, which consists of three WTs and a 0.300 KW solar PV system, is less than the load demand:

$$P\_{W1}(t) + P\_{W2-PV}(t) + P\_{W3}(t) < P\_{Load} \tag{10}$$

#### **5. System Configuration of the HRES System**

In this section, the system configuration of the HRES system is explained. The PV panel is connected with a rectifier to deliver alternating current. The proposed combination of the WT-PV system will act as a power injection system to the power grid and does not have the features needed to apply any sort of power quality control on the generated power. The output power of the WT is directly delivered by means of the AC grid to the distribution side. The output power produced by the solar panels is fed into the grid through the DC-AC converter. A bidirectional charge controller carries out the functions of charging and discharging the battery and thus performing both the AC-DC conversion and voltage boosting. The non-continuous availability of the renewable sources would hinder a system not equipped for gathering the power dispatch standards of the load dispatch centers. Likewise, during periods of high wind, the wind turbines are forced to shut down to safeguard the equipment. Coordinating the battery with the renewable energy source devices will enable to store the excess energy and it could be used beneficially during peak power demand periods, thus improving the system reliability. It additionally enables the storage of energy duringperiods when low demand exists, so that it can be better utilized during the peak load periods. The system configuration of the HRES system in the proposed strategy is shown in Figure 4.

**Figure 4.** System configuration of the HRES system.

Photovoltaic systems and wind turbines are non-dispatchable resources. The battery system is anenergy storage device. In the proposed approach, in the light of the annualized capital and replacement cost the objective functions are resolved. Along these lines, the optimal operation is accomplished when the objective of maximized revenues and minimized usage of HRES is fulfilled.

#### **6. Economic Analysis of Battery Storage System for HRES System Using Hybrid Strategy**

In this section, the proposed strategy for the economic analysis of battery storage system for the HRES systemis described. The proposed strategy is the incorporation of Radial Basis Function Neural Network (RBFNN) and Oppositional Elephant Herding Optimization (OEHO) named RBFNOEHO. RBFNN can be regarded as a feed-forward network composed with three layers of neurons [36]. The EHO algorithm is developed from the natural behavior of elephant herding [37]. Figure 5 portrays the procedure of economic analysis of the proposed system.

**Figure 5.** Process of economic analysis of the proposed system.

In order to accelerate the convergence rate and performance, the EHO is enhanced with the concept of oppositional-based learning (OBL). With the help of RBFNN, the required load demand for the HRES system is continuously tracked. With the consideration of the predicted load demand, OEHO is optimized for the HRES perfect combination. The involvement of the annualized capital cost and replacement cost of the HRES system is the main objective of the proposed methodology. The constraint is the renewable energy sources accessibility, power demand and storage elements state of charge. The systematic process of RBFNN and OEHO are discussedin the following section.

### *6.1. Prediction of Load Demand Using RBFNN*

RBFNN is an artificial neural network. In the field of mathematical modelling, the proposed artificial neural network uses radial basis functions as activation functions. The first layer is called the input layer. It consists of the source nodes for the input data. The second layer is the single hidden layer in the network. It is made up of the radial basis functions. The nonlinear transformation is applied by these functions from the input layer into the hidden layer. The third layer is called the output layer. The output layer of the network is a linear combination of radial basis functions of the inputs and neuron parameters.

Here, the RBFNN is trained by the target power demand with the appropriate input time intervals of the day. To establish the proposed approach, the selection of the input variables to each node in the first layer and the output is computed using the Gaussian membership function.

Based on these corresponding inputs and the output values the RBFNN is trained. The structure and the training process of the ANFIS are describedas follows:

*Step 1: The Input Vector***.** In the input layer of the network, the input vector *a* is applied. In the proposed approach, the time interval *T* is the input of the network, load demand are the output of the network. Then the equation for the input vector is given by:

$$a = \begin{bmatrix} a\_1 \ a\_2 \ \cdots \ a\_p \end{bmatrix}^T \tag{11}$$

Here, *a* represents the input vector of the RBFNN.

*Step 2: The RBF Neurons***.** From the training set, each RBF neuron stores a 'prototype' vector, which is one of the vectors. Each RBF neuron compares the input vector to its prototype, and outputs a value between 0 and 1 which is a measure of similarity. Output of the RBF neuron will be 1 if the input is equal to the prototype. The value of the neuron's response is called its 'activation' value. The prototype vector is called neuron's 'centre'.

*Step 3: The Output Nodes***.** The output of the network consists of set of nodes; each output node computes a score sort for the associated category. Typically, by assigning the input to the category with the highest score a classification decision is made. From every RBF neuron, a score is computed by taking a weighted sum of the activation values. With each of the RBF neurons, an output node associates a weight value by weighted sum and before adding it to the total response, it multiplies the neuron's activation by this weight. Every output node has its own set of weights for a different category because each output node is computing the score. Output node give positive weight to the RBF neurons and negative weight is shared to the others.

*Step 4: RBF Neuron Activation Function*. In between the input and its prototype vector, each RBF neuron computes a measure of the similarity. Input vectors are similar to the prototype, which returns a result closer to 1. There are different possible choices of similarity functions based on Gaussian. The equation for the Gaussian with a one-dimensional input is postulated as follows:

$$f(a) = \frac{1}{a\sqrt{2\pi}}e^{-\frac{\left(a-\beta\right)^2}{2a^2}}\tag{12}$$

Here, input can be represented as *a*, mean as β, standard deviation can be represented as α*.* The RBF neuron activation function is slightly different, and is typically written as:

$$
\psi(a) = e^{-\eta \|a-\beta\| \/ 2} \tag{13}
$$

Here, β is the mean of the distribution in the Gaussian distribution. After completing the algorithm, the RBFNN generates the load demand. Then the predicted values are used as the input of OEHO to ge<sup>t</sup> the optimal energy managemen<sup>t</sup> outputs for the HRES system.

### *6.2. Minimization of Total Cost Using OEHO*

An efficient OEHO approach is presented for the optimal energy managemen<sup>t</sup> in the connected HRES system. EHO algorithm is developed from the natural behavior of elephant herding. An elephant group is composed of a number of clans headed by a matriarch. A clan consists of females and their calves. It is seen that the preference of the females is to always live with family members; whereas, the preference of males is to live a nomadic and solitary life. The improved version of the elephant herding optimization to solve the optimal energy managemen<sup>t</sup> in the HRES system is achieved by introducing oppositional-based learning (OBL). In general, the initial population for all the evolutionary algorithms is generated randomly and gradually until they reached the destination of the 'optimal solution' in the subsequent iterations and stopped at the pre-defined condition. The convergence times of the algorithms are linked to the distances of these initial guesses from the optimal solution. The speed of the convergence and the time taken depend upon the selection of the initial solution. If it is, closer to the optimal solution, then it converges as quickly as possible; otherwise, its speed would be comparatively less and it takes a longer time to converge. The Oppositional-based Learning (OBL) is considered as one of the most efficient concepts. It can be used to improve the initial solution by evaluating both the current candidate solution and its opposite solution simultaneously; and by choosing the more fitted one as the initial solution. OEHO is used to establish the exact schedule of the HRES combinations as per the power variations in both source and load side of the grid. Two definitions are used in the OBL-based proposed system, opposite number and opposite point. The two stages utilized here areOpposition-based Population Initialization (OBPI) and Opposition-based Generation Jumping (OBGJ). The terms are described as follows:

**Definition 1.** *Opposite number: Let the real number be S. The opposite number of s*(*s*<sup>∗</sup>) *is described as follows:*

$$s^\* = a + b - s \tag{14}$$

*Likewise, in Definition 2 this definition can be extended to higher dimensions, which are stated as follows.*

**Definition 2.** *Opposite point: In d-dimensional search space the equation is equated to be S* = *s*1,*s*2,*s*3, ... ,*sD, where s*1,*s*2,*s*3, ... ,*sD* ∈ *Real, real number is represented as Real, si* ∈ [*ai*, *bi*] ∀*i* ∈ {1, 2, ... , *d*}*. The opposite point OS* = *s*<sup>∗</sup>1,*s*<sup>∗</sup>2,*s*<sup>∗</sup>3, ... ,*s*<sup>∗</sup>*D is completely defined by its components:*

$$OS = a\_i + b\_i - s\_i \tag{15}$$

In the accompanying section, the steps of the OEHO algorithm are explained as follows:

*Step 1: Opposition Based Initial Population***.** Opposition based initialize the population of the HRES sources such as WT, PV are taken as the input.

*Step 2: Random Generation***.** The elephant herd population is randomly generated by using the load demand matrix *d*:

$$d = \begin{bmatrix} r\_{11} & r\_{12} & \dots & r\_{1n} \\ r\_{21} & r\_{22} & \dots & r\_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ r\_{n1} & r\_{n2} & \dots & r\_{nn} \end{bmatrix} \tag{16}$$

Here, load demand can be specified as *d*, random solutions can be expressed as *r*.

*Step 3: Fitness Evaluation.* The fitness function is evaluated by minimizing the error value and the equation is expressed in the following equation,

$$obj \; \;= \text{Min} \; \; (\text{Cost}) \tag{17}$$

After the fitness function, the clan updating operator and separating operator is evaluated and the equations are explained in the Step 4 and Step 5.

*Step 4: Clan Updating Operator***.** Being the leaders of the elephant group, the matriarchs are influenced by all the members of a particular clan in each generation; the next position of the member's *j*-th elephant of the clan is calculated as follows:

$$f\_{n\text{new},c\_i,j} = f\_{c\_i,j} + q \times \left(f\_{\text{best},c\_i} - f\_{c\_i,j}\right) \times r \tag{18}$$

Here, the clan is represented as *ci*, the fittest elephant of the clan as *fbest*,*ci* , and the new updated position of the j-th elephant in the clan can be expressed as *fnew*,*ci*,*j*. The next updated position of the fittest elephant is expressed as:

$$f\_{\text{new}, \mathcal{E}\_i, \mathbf{j}} = \gamma \times f\_{\text{center}, \mathcal{E}\_i} \tag{19}$$

$$f\_{catter,c\_i} = \frac{1}{m\_{c\_i}} \times \sum\_{m\_{c\_i}}^{j=1} f\_{c\_i} d\tag{20}$$

Here, the number of elephants of each clan can be represented as *mci* , 1 ≤ *d* ≤ *D* is the *d*-th dimension and *D* is specified as a total dimension.

*Step 5: Separating Operator***.** The worst elephant of each clan updates its position in each generation and the equation can be expressed as follows:

$$f\_{\text{uvrst},c\_i} = f\_{\text{min}} + (f\_{\text{max}} - f\_{\text{min}}) \times random \tag{21}$$

Here, the worst elephant of the clan of elephant group can be indicated as *fworst*,*ci* , maximum and minimum value of the search space can be represented as *f*max and *f*min, *random* ∈ [0, 1] follows the uniform distribution.

*Step 6: Opposition Based Generation Jumping.* To hop to another candidate solution the completely evolutionary process can be constrained in the event that we apply a comparable approach to the

current population, which is more reasonable than the current one. The new population is created and opposite population is determined in the wake of following the EHO operator, dependent on a jumping rate and it is a random number from (0, 1). The fittest NF individuals are chosen from this correlation. To compute the opposite points search space is diminished in every generation:

$$OS\_{\bar{i},\bar{j}} = \min\_{\bar{j}}^{\text{gen}} + \max\_{\bar{j}}^{\text{gen}} - S\_{\bar{i},\bar{j}} \tag{22}$$

Here, *i* = 1,2, ... ., *NF*; *j* = 1,2, ... , *n*, min*gen j* + max*gen j* is the current interval in the population which is becoming increasingly smaller than the corresponding initial range *pj*, *qj*, *JR* as the jumping rate. The process of position updating is continued, until the stop condition is met.

The flowchart of the proposed technique is depicted in Figure 6.

**Figure 6.** Flowchart of the proposed technique.
