Keywords

1 Introduction

Sustainable, renewable, efficient and economical energy systems are the need of the hour for meeting the power demand of increased population. Implementation of microgrid (MG) has gained popularity as a solution to this increased power demand. However, MG has its own challenges for economic operations. Uncertainty in the output of renewable energy sources (RES), energy storage (ES) capacity management, optimization of MG operation with real-time electricity price in market, minimizing operational cost and emissions are some challenges faced when MG is incorporated in the power system [1]. Solutions to these problems like dynamic scheduling of MG using NSGA-II algorithm [2], use of approximate dynamic programming and deep recurrent neural network learning in MG energy management [3], short term generation scheduling [4], scheduling in a CHP-based MG for economic power sharing [5], etc., have evolved to fulfil the interests of all stakeholders in power market.

In recent years, a lot of researchers have been focusing on the operation of MG. Optimal scheduling has always been one of the most important functions in minimizing the net cost of MG [6]. Dynamic optimal scheduling is a good option for MG operation because it considers the lowest cost in scheduling as well as coordinates among different distribution generations (DERs) over many periods.

In India, more than 70% conventional sources of energy are thermal plants which use coal as major fuel. Burning of coal produces harmful gases which degrade our air quality. Also, the price of fuel used is increasing day by day. Under these conditions, sharing of demand by DERs is not only governed by the units’ capability of minimizing the total fuel cost of system generation but also the capability of satisfying the emission requirements. Many optimization algorithms have been used for solving this problem of minimizing fuel cost and emissions. Metaheuristic optimization techniques have gained popularity within last two decades for solution of complex optimization problem. Grey wolf optimization (GWO) [7] is a recently developed metaheuristic technique which is inspired by the hierarchal arrangement in hunting mechanism of grey wolfs.

In this chapter, GWO is used for dynamic scheduling of energy resources considering environmental constraints. Remaining chapters are organized as follows: Problem formulation of this system is given in Sect. 2, the working of the optimization method is described in Sect. 3, results and discussion after using this model are explained in Sect. 4 and the conclusions drawn are compiled in Sect. 5.

2 Problem Formulation

The fuel costs of the conventional generators in a dynamic environment of 24 h which is a convex polynomial can be mathematically expressed as (in $/h) [10]:

$${F\left( P \right) = \mathop \sum \limits_{t = 1}^{24} \mathop \sum \limits_{i = 1}^{6} \left\{ {a_{i} \times P_{i}^{2} \left( t \right) + b_{i} \times P_{i} \left( t \right) + c_{i} } \right\}}$$
(1)

Similarly, emission dispatch function (in Kg/h) is also a convex polynomial and can be written as [10]:

$${E\left( P \right) = \mathop \sum \limits_{t = 1}^{24} \mathop \sum \limits_{i = 1}^{6} \left\{ {\alpha_{i} \times P_{i}^{2} \left( t \right) + \beta_{i} \times P_{i} \left( t \right) + \gamma_{i} } \right\}}$$
(2)

Thus, the multi-objective economic emission dispatch problem can be mathematically stated as [10]:

$$C\left( P \right) = \mathop \sum \limits_{t = 1}^{24} \mathop \sum \limits_{i = 1}^{6} \left[ {\left\{ {a_{i} P_{i}^{2} \left( t \right) + b_{i} P_{i} \left( t \right) + c_{i} } \right\} + {\text{ppf}} \times \left\{ {\alpha_{i} \times P_{i}^{2} \left( t \right) + \beta_{i} \times P_{i} \left( t \right) + \gamma_{i} } \right\} } \right]$$
(3)

where ppf is price penalty factor which is given by

$${\text{ppf}} = \frac{{\left\{ {a_{i} P_{i\hbox{max} }^{2} \left( t \right) + b_{i} P_{i\hbox{max} } \left( t \right) + c_{i} } \right\}}}{{\alpha_{i} \times P_{i}^{2} \left( t \right) + \beta_{i} \times P_{i} \left( t \right) + \gamma_{i} }}$$
(4)

The power generated by each solar PV unit \(\left( {\text{in MW}} \right)\) at t-th hour in a solar farm is given by [11]:

$$P_{\text{gs}} = P_{\text{rated}} \left\{ {1 + \mu \left( {T_{\text{amb}} - T_{\text{ref}} } \right) \times \frac{{S_{t} }}{1000}} \right\}$$
(5)

Cost of operation for the solar farm for 24 h is given as:

$$\mathop \sum \limits_{t = 1}^{24} \mathop \sum \limits_{j = 1}^{13} P_{{{\text{gs}} }} \times C_{j}$$
(6)

The multi-objective cost function of the hybrid system becomes [11]:

$$\begin{aligned} C\left( P \right) & = \mathop \sum \limits_{t = 1}^{24} \left[ {w \times \left( {\mathop \sum \limits_{i = 1}^{6} \left\{ {a_{i} P_{i}^{2} \left( t \right) + b_{i} P_{i} \left( t \right) + c_{i} } \right\}} \right)} \right. + {\text{ppf}} \times \left( {1 - w} \right) \\ & \quad \times \left( {\mathop \sum \limits_{i = 1}^{6} \left\{ {\alpha_{i} \times P_{i}^{2} \left( t \right) + \beta_{i} \times P_{i} \left( t \right) + \gamma_{i} } \right\}} \right) + \left. {\mathop \sum \limits_{j = 1}^{13} P_{\text{gs }} \times C_{j} } \right] \\ \end{aligned}$$
(7)

2.1 Inequality Constraints

The power generated by the conventional thermal plants as well as the RS (Solar PV farm) must lie between maximum and minimum limits. Mathematically,

$$P_{i}^{\hbox{min} } \le P_{i} \le P_{i}^{\hbox{max} }$$
(8)
$$P_{\text{gs}}^{\hbox{min} } \le P_{\text{gs}} \le P_{\text{gs}}^{\hbox{max} }$$
(9)

The ramp rate limits for thermal unit power generation are considered in this problem. The power generation of thermal units is constrained by the ramp rate limits as follows:

$$P_{i}^{t} - P_{i}^{t - 1} \le {\text{UR}}_{i}$$
(10)
$$P_{i}^{t - 1} - P_{i}^{t} \le {\text{DR}}_{i}$$
(11)

2.2 Equality Constraints

The power generated at any instant of time by all the thermal plants and the RS (Solar PV farm) should satisfy the total desired load of the system which is mathematically described as:

$$P_{\text{Load}} = \mathop \sum \limits_{i = }^{6} P_{i} + \mathop \sum \limits_{j = 1}^{13} P_{\text{gs}} + P_{L}$$
(12)

3 Grey Wolf Optimization

Grey wolf optimization (GWO) is belonging to the family of swarm intelligence [7]. Its analytical model mimics the intelligent, self-organized group behaviour of grey wolves for hunting prey in nature. Grey wolves live in a group of 5–15 members. They follow a proper hierarchy with four types of member represented as Alpha, Beta, Delta and Omega. The social hierarchy of grey wolves is illustrated in Fig. 1. Systematic organization and discipline are their main strength.

Fig. 1
figure 1

Social hierarchy of grey wolves in nature

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Group leader is male/female represented by Alpha. He or She is only the decision maker for hunting, walking and selection of place for sleeping. Beta wolf has second place in social hierarchy and helps group leader in decision making. Delta is the subordinates of alpha and beta but they dominate over omega. Delta has four subgroups: Scouts, Sentinels, Hunters and Caretakers. Scouts are responsible for watching boundary territory and warning the group members in case of any danger. Sentinels are responsible for the protection of group members. Hunters help alpha and beta in hunting and also responsible for arranging the food for the group members. Weak and wounded member are taken care by caretakers. Omega plays the role of scapegoat in the group and they generally eat at last only.

On the basis of above-disciplined group behaviour, the analytical model of GWO is described by three phases during hunting which are described as below.

  1. (a)

    Entrapment of prey

In its first phase, model is based upon assumption that grey wolves update their position one with respect to other in n-dimensional search space as below [7].

$${\mathcal{D}} = \left| {{\mathcal{C}} \times {\mathcal{X}}_{P} \left( t \right) - {\mathcal{X}}\left( t \right)} \right|$$
(13)
$${\mathcal{X}}\left( {t + 1} \right) = {\mathcal{X}}_{P} \left( t \right) - {\mathcal{A}} \times {\mathcal{D}}$$
(14)
$${\mathcal{A}} = 2ar_{1} - a$$
(15)
$${\mathcal{C}} = 2r_{2}$$
(16)
$$a = 2 - \left( t \right)\left( {\frac{2}{T}} \right)$$
(17)

The value ‘a’ is linearly decreased from 2 to 0 over the course of iterations and Fig. 2 illustrates this phase

Fig. 2
figure 2

Entrapment of prey phase

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  1. (b)

    Hunting of Prey

In order to simulate self-organized and group behaviour of grey wolves, alpha, beta and gamma are considered as three best solutions. Alpha is assumed to be closest to the best solution followed by the solution of beta and gamma. Therefore, during optimization process, first three solutions are considered as the best and remainders are considered as omega. The position is updated with respect to the position of omega. The position of omega \(\left( \omega \right)\) will vary as per the current best position in algorithm. The final position is defined with respect to position of alpha, beta and delta in search space as below.

$${\mathcal{D}}_{\alpha } = \left| {{\mathcal{C}}_{1} .{\mathcal{X}}_{\alpha } - {\mathcal{X}}} \right|, \;{\mathcal{D}}_{\beta } = \left| {{\mathcal{C}}_{2} .{\mathcal{X}}_{\beta } - {\mathcal{X}}} \right| ,\; {\mathcal{D}}_{\delta } = \left| {{\mathcal{C}}_{3} .{\mathcal{X}}_{\delta } - {\mathcal{X}}} \right|$$
(18)
$${\mathcal{X}}_{1} = {\mathcal{X}}_{\alpha } \left( t \right) - {\mathcal{A}}_{1} \times {\mathcal{D}}_{\alpha } , {\mathcal{X}}_{2} = {\mathcal{X}}_{\beta } \left( t \right) - {\mathcal{A}}_{2} \times {\mathcal{D}}_{\beta } , {\mathcal{X}}_{3} = {\mathcal{X}}_{\delta } \left( t \right) - {\mathcal{A}}_{3} \times {\mathcal{D}}_{\delta }$$
(19)
$${\mathcal{X}}\left( {t + 1} \right) = \frac{1}{3} \times \left( {{\mathcal{X}}_{1} + {\mathcal{X}}_{2} + {\mathcal{X}}_{3} } \right)$$
(20)
  1. (c)

    Attacking the Prey

In the last stage, grey wolf attacks the prey. In the analytical model, it can be realized by shrinking value of “a” from 2 to 0 as iteration progresses and hence \({\mathcal{A}}\) reduces. The last stage in hunting is attacking the prey when the prey has stopped. This can be achieved mathematically by reducing the value of a gradually from 2 to 0, consequently, \({\mathcal{A}}\) is varied randomly in range [−1, 1].

4 Results and Discussion

The main objective of this chapter is to find the impact of renewable integration on operating cost of fuel and quantity of emissions released, which is discussed in two cases. First case involving only thermal units and second case is a hybrid arrangement of thermal plants with solar PV integration.

4.1 Description of Test Cases

Case 1

This test system contains six thermal power units; its fuel cost, minimum and maximum power limits and emission coefficients which are adapted from [10] and listed in Table 1.

Table 1 Data related to six conventional thermal power plants
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Case 2

It is a hybrid test case having six thermal units similar to Case 1 and a solar PV farm comprising of 13 PV units. The required data of the solar PV farm are adapted from [11] and illustrated in Fig. 3 and listed in Table 2. Figure 4 provides the data of temperature (°C) and solar radiation (W/m2) of PV on a single day for 24 h. Table 2 gives data of rated power and per unit cost of thirteen PV units in the solar farm.

Fig. 3
figure 3

Solar PV data of temperature (°C) and radiation (W/m2)

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Table 2 Rated power and per unit price of solar PV units in the solar farm
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Fig. 4
figure 4

Statistical and computational comparison of Case 1

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4.2 Simulation Results

GWO is implemented for solution of ELD, EED and CEED problem in MATLAB R2013a environment. For each case, GWO algorithm was run for 30 times and best results are tabulated in Tables 3, 4, 5, 6 and 7.

Table 3 Comparison of cost and emissions for different algorithms (Case 1)
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Table 4 Optimum generation schedule (CEED) obtained by GWO (Case 1)
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Table 5 Parameters used for different algorithms
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Table 6 Comparison of cost and emissions for different algorithms (Case 2)
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Table 7 Optimum generation schedule (CEED) obtained by GWO (Case 2)
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The performance of GWO with recent methods like artificial bee colony (ABC) [8] and differential evolution (DE) [9] is given in Table 3. Table 4 tabulates the optimum scheduling of the six thermal units for CEED. The parameters considered in implementing the algorithms are given in Table 5. Here, it is observed that the optimum results in terms of minimum cost and least emissions obtained by GWO are lowest as compared to the results obtained by simulation using ABC [8] and DE [9]. The statistical comparison in Fig. 4 illustrates that though the average CPU time in computation is more for GWO than ABC and DE, the standard deviation obtained in results by GWO is lowest than the other two methods.

The optimal solution in terms of cost and emission for hybrid thermal–PV system is listed in Table 6. By comparing results, it can be observed that the total cost for hybrid system is found to be lowest for GWO as compared to other two metaheuristic methods for all three objective functions taken into consideration. The optimal generation scheduled for CEED obtained using GWO is tabulated in Table 7. Here, it is observed that all associated operational constraints (8)–(11) are fully satisfied.

5 Conclusion

This chapter focuses on using recently evolved nature-inspired technique named as grey wolf optimization (GWO) for solution of a hybrid thermal–PV system working as power producers in a microgrid in island mode. After analysing the illustrations above, it can be concluded that GWO provides better results as compared to two other well-proven optimization techniques which are ABC and DE. In dynamic environment, the GWO algorithm converged in an efficient manner for solution of environmental/economic dispatch problem in dynamic environment without violating any constraint.

In Case 1, GWO optimizes the minimal cost (ELD) and gives least emissions (EED) as compared to ABC and DE. In Case 2, the microgrid using thermal–PV units as DERs have lesser cost of operation, lower fuel cost and lesser emissions than in Case 1. Thus, using renewable sources of energy will economically and ecologically make the existing microgrid more efficient.

Microgrid using the proposed hybrid thermal–PV system implementing GWO as optimization methodology will be an economic and efficient way to solve the modern-day multi-objective power scheduling problems.