Optimal generation scheduling and dispatch of thermal generating units considering impact of wind penetration using hGWO-RES algorithm

Abstract

In order to achieve paramount economy, hybrid renewable energy sources are gaining importance, as renewable sources are costless. Over the past few years wind energy incorporation drew more consideration in the electricity market, as wind power took an affirmative role in energy saving as well as sinking emission pollutants. Recently developed Grey wolf optimizer (GWO) algorithm has conspicuous behavior for verdicting global optima, without getting ensnared in premature convergence. In the proposed research the exploitation phase of the grey wolf optimizer has been further improved using random exploratory search algorithm, which uses perturbed solutions vectors along with previously generated solution vectors. The paper presents a hybrid version of Grey Wolf Optimizer algorithm combined with random exploratory search algorithm (hGWO-RES) for the solution of combinatorial scheduling and dispatch problem of electric power systems. To validate the feasibility of the algorithm, the proposed algorithm has been tested on 23 benchmark problems. To verify the feasibility and efficacy of operation of proposed algorithm on generation scheduling and dispatch of electric power systems, small and medium scale power systems consisting of 7-, 10-, 19-, 20- and 40-generating units systems taken into consideration. Commitment and scheduling pattern has been evaluated with and without wind integration and it has been experimentally founded that proposed hybrid algorithm provides superior solution as compared to other recently reported meta-heuristics search algorithms.

1 Introduction

Hybrid renewable energy sources are getting importance, as renewable sources are costless. Over the past few years, wind energy incorporation drew more consideration in electricity generation market, as wind power acting as an affirmative role in energy saving as well as sinking emission pollutants. Also, multidisciplinary design optimization and multidisciplinary system design optimization are emerging area for the solution of design and optimization problems incorporating a number of disciplines. Scientific revolution has affected every aspect of contemporary life. In recent years, with the advancement in computer technology, new era of problem-solving methods has been emerged making use of computers. These methods are becoming more suitable for solving complex problems. These problem-solving methods with direct human involvement are sluggish. So, computer-aided design are widely adopted emphasizing on use of computer for engineering design problems. The computer-aided design not only emphasis on simulating a system but also helps to find the optimal design with high accuracy, low cost, high speed and reliability. Optimization techniques are considered to be one of the best tools for solving the engineering problems and to find the optimal results for the problem. The optimization process initialize with random set for specified problem and then improving them over predefined steps. The engineering problems to be tackled consist of various difficulties such as unconstrained, constrained, uncertainties, local solution, global solution, multiple objectives, etc. Optimization technique must be able to discourse these issues. In the recent years, various meta-heuristics search algorithms has been implemented such as Biogeography based Optimizer [1], Grey Wolf Optimizer [2], Ant Lion Optimizer [3], Moth Flame Optimizer [4], Multi Verse Optimizer [5], Dragon Fly Algorithm [6], Sine Cosine Algorithm [7], Lightning Search Algorithm [8], Seeker Optimization Algorithm [9],Virus Colony Search Algorithm [10], Whale Optimization Algorithm [11], Wind Driven Optimization [12], Water Cycle Algorithm [13], Salp Swarm Algorithm [14], Symbiotic Organism Search [15], Search Group Algorithm [16], Stochastic Fractal Search Algorithm [17], The Runner Root Algorithm [18], Ant Colony Optimization [19], Shuffled Frog Leaping Algorithm [20], Flower Pollination Algorithm [21], Optics Inspired Optimization [22], Cultural Evolution Algorithm [23], Grasshopper Optimization Algorithm [24], Interior Search Algorithm [25], Colliding Bodies Optimization [26], Krill Herd Algorithm [27], Competition over Resources [28], Binary Bat Algorithm [29], Mine Blast Algorithm [30], Biogeography Based Optimization [31], Adaptive Cuckoo Search Algorithm [32], Bat Algorithm [33], Animal Migration Optimization [34], Gravitational Search Algorithm [35], Branch and Bound Method [36], Expert System Algorithm [37], Genetic Algorithm [38], Binary Gravitational Search Algorithm [39], Collective Animal Behavior Algorithm [40], Bird Swarm Algorithm [41], Cognitive Behavior Optimization [42], Electromagnetic Field Optimization [43], Firework Algorithm [44], Water Wave Optimization [45], Earthworm Optimization Algorithm [46], Forest Optimization Algorithm [47], Mean Variance Optimization Algorithm [48], League Championship Algorithm [49], Chaotic Krill Herd Algorithm [50], Elephant Herding Optimization [51], Differential Evolution Algorithm [52], Imperialistic competition algorithm [53], Invasive weed optimization [54], Particle swarm optimization algorithm [55], Crow Search Algorithm [56], Self-Adaptive Bat Algorithm [57]. The brief review of these algorithms is depicted in Table 1. A large portion of these calculation depends on straight and nonlinear programming systems that require broad slope data and for the most part attempt to locate an enhanced arrangement in the region of a beginning stage. These numerical improvement calculations give a valuable technique the worldwide ideal in basic and perfect models [3]. However, some certifiable designing and logical improvement issues are exceptionally intricate and hard to settle, utilizing these techniques. On the off chance that there are more than one neighborhood minima in the problem, the outcome may rely upon the choice of an underlying point, and the acquired minima may not really be the global minima. Also, The No-Free-Lunch theorem for optimization allow developers to develop a new algorithm or to improve the existing algorithm because, it logically proves that there is no such optimization algorithm which can solve all the optimization problems with equal efficiency for all. Some algorithms work best for a few problems and worst for the rest of the problems. So, there is always a scope or improvement to develop the algorithm which could work well for most of the problems. In the proposed research, hybrid variant of grey wolf optimizer has been implemented to solve combinatorial optimizations problems of multidisciplinary system design.

Table 1 Brief review of various heuristics and meta-heuristics search algorithms

1.1 Novely and contribution

The main contributions of the proposed research study are as follows:

• A novel hybrid grey wolf inspired optimizer algorithm has been proposed by improving exploitation phase of existing GWO algorithm using random exploratory search algorithm.

• In the proposed research, two algorithms are combined recursively to improve the local search capability of existing GWO algorithm and proposed algorithm has been tested for 23-benchmark problems including unimodal, multimodal and fixed dimension optimization.

• Performance analysis of the proposed algorithm has been investigated for standard Numerical Optimization Problem.

• For validation, the proposed algorithm has been tested on combinatorial optimization problem (i.e. Unit Commitment and dispatch Problem) of electric power system for small and medium scale power systems consisting of 7-, 10-, 19-, 20- and 40-generaing units.

• The performance of the proposed algorithm has been investigated by comparing it with various recently developed meta-heuristics search algorithm.

2 Unit commitment problem formulation

Unit Commitment of power system units is a multidimensional optimization task for preparation and maneuver of participated units. Contemporary power system network has diverse generating resources, which can be broadly grouped together into two categories i.e. conventional and non-conventional generation sources. Unfortunately, load demand is never steady and it has the tendency to change at every instant of time, a great difficulty arises for the generation that tends to cope with this variable load. Thus, it is required to make a judgment on which generating unit to turn on and which unit to turnoff and at what time it is desirable in the power system network. This complex process of obtaining on-off pattern of generating units, which should satisfy the load demand and spinning reserve parameter is known as unit commitment problem [1]. Unit commitment problem is a part of system planning schedule of 8-h to 24 h planning is done before-hand this duration is quite moderate, however it can to lead short-term planning (next hour) to very long-term planning (one week to few weeks). Basically, unit commitment problem is a hierarchical problem as it does not end with the achievement of bare on-off patterns of units but economic factors are deeply incorporated with it. Next level of problem is allocation of real power in units that participated in load. So this problem can be subdivided into two sub problem Viz. optimum allocation (commitment) of generators at each stations for various load levels and “allocation of generation” to each station. The first problem in power system dialect is called the unit commitment problem (UCP) and second is called load scheduling or dispatch problem (EDP). Since this problem has both binary (UC) as well as continuous (ED) variable, it is known as the link optimization problem. In recent years, due to tremendous increase in load demand, large interconnections of hybrid electric networks are taken into consideration, which basically consist of an integration of thermal unit with one renewable energy source as a wind system, acknowledged as hybrid renewable energy system (HRES) [64,65,66]. The vagueness in Wind Power crafts difficulties for obtaining UC Patterns. Wind integrated thermal power systems are analyzed on the basis of various simulation techniques such as Weibull probability distribution function [67], diverse Probability distribution function [68], adaptable Probability distribution function [69], incomplete Gama function [70], artificial neural networks [71], adaptive neuro-fuzzy approach [71], Gaussian PDF [72], copula theory [73], Levy alpha-stable distribution function [74], P-SCOPF [75], Differential evolution [76], Genetic algorithm [77], hPSO-SQP [78, 79], Upgraded Inertia Wight (PSO-IIW) [80], Fuzzy approach [81], neural network with MIPSO algorithm [82] and Teaching learning based optimizer [83]. No free lunch theorem has logically demonstrated that there exists no method suitable to all optimization problems [84]. Hence, the hybrid variant of grey wolf optimizer combined with random exploratory search algorithm has been proposed to evaluate the generation scheduling and dispatch of thermal power system combined with renewable energy system. The objective function for thermal power system with consideration of wind power can be mathematically described as per Eq.(1), as wind turbine do not consume fossil fuel and does not include any fuel cost.

$$ F{C}_T=\sum \limits_{h=1}^H\sum \limits_{n=1}^{NOU}F{C}_n\left({P}_n^h\right){U}_n^h+{U}_n^h\left(1-{U}_n^{h-1}\right) SU{C}_{n,h}+{U}_n^{h-1}\left(1-{U}_n^h\right) SD{C}_n $$

(1)

where, $F{C}_n\left({P}_n^h\right)$ describe the fuel cost of n-th generating units at h-th hours and SUC_{n, h} represents the startup cost of n-th generating units for h-th hours and these costs may be mathematically described as:

$$ F{C}_n\left({P}_n^h\right)={a}_n{\left({P}_n^h\right)}^2+{b}_n\left({P}_n^h\right)+{c}_n $$

(2)

$$ SU{C}_{n,h}=\left\{\begin{array}{l} HS{C}_n\\ {} CS{C}_n\end{array}\kern0.5em \begin{array}{l} if\\ {} if\end{array}\kern0.5em \begin{array}{l}{T}_{n, down}\le {T}_{n, off}^h\le {T}_{n, down}+{T}_{n, cold}\\ {}{T}_{n, off}^h\ge {T}_{n, down}+{T}_{n, cold}\end{array}\right\} $$

(3)

Where, HSC_n hot start is cost, and CSC_n is cold start cost, T_{n, down} is minimum down time of n-th unit, ${T}_{n, off}^h$ is consecutive off time of n-th unit and term T_{n, cold} represents the cold start hour of the n-th units.

The aforementioned unit commitment problem is subjected to various equality and non-equality constraints and which are mathematically described below:

• a) Power Operational constraints:

$$ \sum \limits_{i=1}^N{P}_{i,t}+{P}_{W,t}-{P}_{D,t}=0 $$

(4)

• b) Spinning Reserve Constraint

$$ S{R}_{j,u}^h=\min \left({P}_{j,\max }-{P}_{j,h},\kern0.5em {U}_{R,h}{T}_l\right) $$

(5)

$$ \sum \limits_{n=1}^{NOU}{u}_{n,h}S{R}_{n,h}^h\ge {R}_D^h+{W}_u.{P}_{w,h} $$

(6)

• c) Minimum up and down time constraints

$$ \left({P}_{n,h-1}^{on}-{P}_{n,\min}^{on}\right)\left({U}_{n,h-1}-{U}_{n,h}\right)\ge 0 $$

(7)

$$ \left({T}_{j,t-1}^{off}-{T}_{j,\min}^{off}\right)\left({U}_{j,t-1}-{U}_{i,t}\right)\ge 0 $$

(8)

• d) Maximum and Minimum Power Limit

$$ {P}_n^{\mathrm{min}}\le {P}_{n,h}\le {P}_n^{\mathrm{max}} $$

(9)

The probability distribution function for the calculation of wind power can be mathematically represented [25].:

$$ pdf\left(v;k,\lambda \right)=\frac{k}{\lambda }{\left(\frac{v}{\lambda}\right)}^{k-1}\exp \left[-{\left(\frac{v}{k}\right)}^k\right] $$

(10)

Because of intermittent nature of wind power (WP), it is a Random variable. The mathematical function for wind power output and Wind speed can be mathematically described as:

$$ {P}_W=\Big\{{\displaystyle \begin{array}{ll}0& \left({v}^h\le {v}_{in}\; or\;{v}^h\ge {v}_{out}\right)\\ {}{P}_{WR}& \left({v}_r\le {v}^h\le {v}_{out}\right)\\ {}\frac{\left(v- vin\right){P}_{WR}}{v_r-{v}_{in}}& \left({v}_{in}\le {v}^h\le {v}_r\right)\end{array}} $$

(11)

For wind speed (v^h), between 0 and v_in or for wind speed greater that v_out the WP is zero. When wind speed (v^h) is between v_r and v_out Wind power is equal to the rated wind power. So for the first and second eventuality in Eq.(11), the Wind power is a discrete variable. The probability of Wind power being 0 or P_WR be calculated as per Eqs.(12) and (13) respectively described below:

$$ {\mathrm{P}}_r\left({P}_W=0\right)= cdf\left({v}_{in}\right)+\left[1- cdf\left({v}_{out}\right)\right] $$

(12)

$$ \mathrm{For}\;{P}_W=0,\kern0.5em {\mathrm{P}}_r=1-\exp \left[-{\left(\frac{v_{in}}{\lambda}\right)}^k\right]+\exp \left[-{\left(\frac{v_{out}}{\lambda}\right)}^k\right] $$

(13)

$$ \mathrm{For}\;{P}_W={P}_{WR},\kern0.5em {P}_r=\exp \left[-{\left(\frac{v_r}{\lambda}\right)}^k\right]-\exp \left[-{\left(\frac{v_{out}}{\lambda}\right)}^k\right] $$

(14)

Between v_in and v_r Wind power is continuous variable and its probability density function can be written as

$$ pdf\left({P}_W\right)=\frac{kL{v}_{in}}{\left({P}_{WR}\right)\lambda}\left[\frac{\left(1+\left(\raisebox{1ex}{$L{P}_W$}\!\left/ \!\raisebox{-1ex}{${P}_{WR}$}\right.\right)\right){v}_{in}}{\lambda}\right]\times \exp \left[-{\left(\frac{1+\left(\raisebox{1ex}{$L{P}_W$}\!\left/ \!\raisebox{-1ex}{${P}_{WR}$}\right.\right) vin}{\lambda}\right)}^k\right] $$

(15)

Ever since Correctness of Wind Power prediction is not meticulous and exact so based on above empirical formulas, Maximum generation scheduling can be computed. This maximum wind generation takes up base portion of load curve. In a time when the present position of unit commitment does not meet with the reserve demand a unit may be startup subject to satisfying constraint as described in Eq. (15).

3 Hybrid grey Wolf optimizer

Primarily developed Grey Wolf Optimizer, is a transformative calculation algorithm, based on grey wolves, which recreate the social stratum and chasing component of grey wolves in view of three principle ventures of chasing: scanning for prey, encompassing prey and assaulting prey and its mathematical model was designed in view point of hierarchy levels of different wolves. The fittest solution was designated as alpha (α). Accordingly, the second and third best solutions are named beta (β) and delta (δ) individually. Whatever is left of the hopeful solution are thought to be omega (ω), kappa (κ) and lambda (λ). For the fitness value calculation, the advancement (i.e. chasing) is guided by α, β and δ. The ω, κ and λ wolves trail these three wolves. In GWO, Encircling or Trapping of Prey was achieved by calculating $\overrightarrow{\mathrm{D}}$ and ${\overrightarrow{\mathrm{X}}}_{\mathrm{GWolf}}$ vectors described by Eqs. (16.1) and (16.2).

$$ \overrightarrow{\mathrm{D}}=\mid \overrightarrow{\mathrm{C}}.{\overrightarrow{\mathrm{X}}}_{\mathrm{Prey}}\left(\mathrm{iter}\right)-{\overrightarrow{\mathrm{X}}}_{\mathrm{GWolf}}\left(\mathrm{iter}\right)\mid $$

(16.1)

$$ {\overrightarrow{\mathrm{X}}}_{\mathrm{GWolf}}\left(\mathrm{iter}+1\right)={\overrightarrow{\mathrm{X}}}_{\mathrm{Prey}}\left(\mathrm{iter}\right)-\overrightarrow{\mathrm{A}}.\overrightarrow{\mathrm{D}} $$

(16.2)

Where, iter demonstrates the present iteration, $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{C}}$ are coefficient vectors, ${\overrightarrow{\mathrm{X}}}_{\mathrm{Prey}}$ is the position vector of the prey and ${\overrightarrow{\mathrm{X}}}_{\mathrm{GWolf}}$ shows the position vector of a grey wolf and the vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{C}}$ are calculated as follows:

$$ \overrightarrow{\mathrm{A}}=2\overrightarrow{\mathrm{a}}.{\overrightarrow{\upmu}}_1-\overrightarrow{\mathrm{a}} $$

(16.3)

$$ \overrightarrow{\mathrm{C}}=2.{\overrightarrow{\upmu}}_2 $$

(16.4)

Where, ${\overrightarrow{\upmu}}_1,{\overrightarrow{\upmu}}_2\in \operatorname{rand}\left(0,1\right)$ and $\overrightarrow{\mathrm{a}}$ decreases linearly from 2 to 0.

The hunting of prey are achieved by calculating the corresponding fitness score and positions of alpha, beta and delta wolves using Eqs. (17), (18) and (19) respectively and final position for attacking towards the prey was calculated by Eq. (20).

$$ {\overrightarrow{\mathrm{D}}}_{\mathrm{Alpha}}=\mathrm{abs}\left({\overrightarrow{\mathrm{C}}}_1.{\overrightarrow{\mathrm{X}}}_{\mathrm{Alpha}}-\overrightarrow{\mathrm{X}}\right) $$

(17a)

$$ {\overrightarrow{\mathrm{X}}}_1={\overrightarrow{\mathrm{X}}}_{\mathrm{A}\mathrm{lpha}}-{\overrightarrow{\mathrm{A}}}_1.{\overrightarrow{\mathrm{D}}}_{\mathrm{A}\mathrm{lpha}} $$

(17b)

$$ {\overrightarrow{\mathrm{D}}}_{\mathrm{Beta}}=\mathrm{abs}\left({\overrightarrow{\mathrm{C}}}_2.{\overrightarrow{\mathrm{X}}}_{\mathrm{Beta}}-\overrightarrow{\mathrm{X}}\right) $$

(18a)

$$ {\overrightarrow{\mathrm{X}}}_2={\overrightarrow{\mathrm{X}}}_{\mathrm{Beta}}-{\overrightarrow{\mathrm{A}}}_2.{\overrightarrow{\mathrm{D}}}_{\mathrm{Beta}} $$

(18b)

$$ {\overrightarrow{\mathrm{D}}}_{\mathrm{D}\mathrm{elta}}=\mathrm{ab}s\left({\overrightarrow{\mathrm{C}}}_3.{\overrightarrow{\mathrm{X}}}_{\mathrm{D}\mathrm{elta}}-\overrightarrow{\mathrm{X}}\right) $$

(19a)

$$ {\overrightarrow{\mathrm{X}}}_3={\overrightarrow{\mathrm{X}}}_{\mathrm{D}\mathrm{elta}}-{\overrightarrow{\mathrm{A}}}_3.{\overrightarrow{\mathrm{D}}}_{\mathrm{D}\mathrm{elta}} $$

(19b)

$$ \overrightarrow{\mathrm{X}}\left(\mathrm{iter}+1\right)=\frac{\left({\overrightarrow{\mathrm{X}}}_1+{\overrightarrow{\mathrm{X}}}_2+{\overrightarrow{\mathrm{X}}}_3\right)}{3} $$

(20)

In the proposed hybrid Grey-Wolf Optimizer-Random Exploratory search (hGWO-RES) algorithm, the position vector ${\overrightarrow{\mathrm{X}}}_{\mathrm{i}}$ is perturbed by Δ_i and new position vectors $\left({\overrightarrow{\mathrm{X}}}_i+{\varDelta}_i\right)$ and $\left({\overrightarrow{\mathrm{X}}}_i-{\varDelta}_i\right)$ has been obtained. The variation of parameter Δ_i has been taken randomly within local search space to exploit the search space in a better way. The new fitness solutions f⁺ ← f (X + Δ) and f⁻ ← f (X − Δ) has been obtained along with previous fitness solution f ← f (X) and final fitness has been evaluated taking minimum values out of these newly obtained solutions using Eq. (21).

$$ {f}_{final}\leftarrow \min \left({f}^{+},{f}^{-},f\right) $$

(21)

The impact of newly obtained positions vectors as 2-Dimentional position vectors and conceivable neighbors are outlined in Fig. 1. As per Fig. 1, a grey wolf poser of (X, Y) can update its position w.r.t. newly obtained position vectors ((X + Δ), (Y + Δ)) and ((X − Δ), (Y − Δ)) indicated by the position of the prey (X*, Y*) and exploit the search space in better way. Better places around as well as can be expected regarding the present position by altering the estimation of $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{C}}$ vectors. Figure 1 shows the 2-D view of Position Vectors along with perturbed position vectors $\left({\overrightarrow{\mathrm{X}}}_i+{\varDelta}_i\right)$, $\left({\overrightarrow{\mathrm{X}}}_i-{\varDelta}_i\right)$ and possible next Location w.r.t. Prey. The 3-D view of position vectors along with perturbed position vectors $\left({\overrightarrow{\mathrm{X}}}_i+{\varDelta}_i\right)$, $\left({\overrightarrow{\mathrm{X}}}_i-{\varDelta}_i\right)$ and possible next location w.r.t. prey has been shown in Fig. 2.

Fig. 1

2-D view of position vectors along with perturbed position vectors $\left({\overrightarrow{\mathrm{X}}}_i+{\varDelta}_i\right)$, $\left({\overrightarrow{\mathrm{X}}}_i-{\varDelta}_i\right)$ and possible next location w.r.t. Prey

Fig. 2

3-D view of position vectors along with perturbed position vectors $\left({\overrightarrow{\mathrm{X}}}_i+{\varDelta}_i\right)$, $\left({\overrightarrow{\mathrm{X}}}_i-{\varDelta}_i\right)$ and possible next location w.r.t. Prey

The random positions vectors, which allow grey wolves to reach any position between the points including perturbed position vectors $\left({\overrightarrow{\mathrm{X}}}_i+{\varDelta}_i\right)$ and $\left({\overrightarrow{\mathrm{X}}}_i-{\varDelta}_i\right)$ are shown in Fig. 3a.

Fig. 3

a Updating of position of alpha, beta and delta Grey Wolves in GWO. b PSEUDO code for proposed hybrid GWO-RES Algorithm

The exploration phase in hGWO-RES is similar to GWO., In order to explore the search space globally, vector $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{C}}$ are used, which mathematically model divergence. The PSEUDO code for the proposed hybrid GWO-RES algorithm is shown in Fig. 3b.

4 Solution strategy for unit commitment problem

In grey wolf optimizer, the search agent explore and exploit their updated position to a suitable real value in given search space considering various constraints impose upon them. Since unit commitment problem is highly constrained in nature, they have both binary and discrete values. Thus mapping of continuous value of search agent updated to binary value is mandatory. Before solving unit commitment problem by using hGWO-RES algorithm we represent agent as a binary string .each unit “on state” as 1 and “off state” as a 0. So, unit state U is basically matrix of {N*T} following steps clarify modus operandi of unit commitment problem.

• Step-1: To solve single area unit commitment problem, every individual is defined as units ON/OFF status showed as 1/0 correspondingly. An individual would display the unit commitment schedule over the time horizon H. The on/off schedule of the units is stored as an integer-matrix U, which is mathematically defined as:

$$ {U}_{NP}=\left[\begin{array}{l}{u}_1^1\\ {}{u}_2^1\\ {}\vdots \\ {}{u}_G^1\end{array}\kern0.5em \begin{array}{l}{u}_1^2\\ {}{u}_2^2\\ {}\vdots \\ {}{u}_G^2\end{array}\kern0.5em \begin{array}{l}\cdots \\ {}\cdots \\ {}\vdots \\ {}\cdots \end{array}\kern0.5em \begin{array}{l}{u}_1^H\\ {}{u}_2^H\\ {}\vdots \\ {}{u}_G^H\end{array}\right], $$

Where, ${u}_n^h$ is unit on/off status of n^th unit at h^th hour (i.e. ${u}_n^h$=1/0 for ON/OFF).

• Step-2: Generating units are prioritized according to their Average Full Load generation Capacity in descending order.

• Step-3: Status of individual units is modified in the population to satisfy the spinning reserve constraints

• Step-4: Individual units in the population are repaired for minimum up/down time violations

• Step-5: Units of some search agents are de-committed in the population to reduce excessive spinning reserve due to minimum up/down time repairing

• Step-6: Economic Load Dispatch Problem is then solved using MIQP and Fuel Cost is calculated for each Hour.

• Step-7: Calculate Start-up cost for each hour using Eq. (3).

• Step-8: Overall generation cost for 1st position is evaluated and it is assumed as global fitness and its position as global position.

• Step-9: Overall generation costs for all positions are then evaluated in the population and then local generation cost and local commitment schedule for whole population is determined.

• Step-10: Overall global generation cost is compared with Local generation cost in whole population. If global generation cost is greater than local generation cost, replace global generation cost with local generation cost and take local commitment schedule as global commitment.

• Step-11: Modify the individual position using hGWO-RES algorithm and determine overall best generation cost and commitment schedule.

• Step-12: If the maximum iteration number is reached, then go to next step (Step 14.)

• Step-13: Otherwise, increase iteration number and go back to step 3.

• Step-14: Stop and obtain the optimal solution of single area unit commitment problem from the individual position in the population that generated the least total generation cost (Fig. 4).

Fig. 4

Flow chart for solution of UCP using hGWO-RES

5 Constraints handling strategy/ repair mechanism of constraints

The achieved major unit scheduling by hGWO-RES may not fulfill the certain crucial constraints such as MDT, MUT, Spinning reserve etc. So, the constraints defilements are to be repaired. In this paper a heuristic search strategy is adopted to tackle such problem.

5.1 Minimum up and minimum down time handling strategy

Minimum up and down time of specific unit is defined as connective hours that unit is ‘on’ or ‘off’ ‘when it ‘on’ or ‘off’. Any unit that is ‘on‘ should not be turned ‘off’ immediately without reaching to ‘MUT’ and similarly any unit which is once “off” should not be turned “on” immediately without reaching to MDT. These constraints are calculated beforehand by using following recursive relation

$$ {T}_{n. on}^h=\Big\{{\displaystyle \begin{array}{l}{T}_{n, on}^{h-1}+1\kern0.5em if\begin{array}{c}{u}_n^h=1\end{array}\\ {}0\begin{array}{cc}\begin{array}{cc}\begin{array}{c}\end{array}& \end{array}& \begin{array}{c}\begin{array}{c}\begin{array}{c}\end{array}\end{array}\begin{array}{c}\begin{array}{c} if\begin{array}{c}{u}_n^h=0\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}} $$

(26)

$$ {T}_{n, off}^h=\Big\{{\displaystyle \begin{array}{l}{T}_{n, off}^{h-1}+1\kern0.5em if\begin{array}{c}{u}_n^h=0\end{array}\\ {}0\begin{array}{cc}& \begin{array}{cc}& \begin{array}{c} if\begin{array}{c}{u}_n^h=1\end{array}\end{array}\end{array}\end{array}\end{array}} $$

(27)

Where ${T}_{n, on}^h$ and ${T}_{n, off}^h$ are number of continuous time when unit is on and off.

When crowning load duration appreciably inferior to the minimum down time of a particular unit. Minimum up time is violated. And constraint associated with minimum down time is violated at low load level where low load duration is considerably shorter than minimum up time. Since repapering of MDT, MUT, can lead to excessive spinning reserve, which results into high operating cost, thus if this remains it would defeat the whole purpose of optimizing cost. Hence we again us heuristic technique to de-commit excess of reserve.

The methodology to adjust/repair defilement of constraints associated with MDT, MUT are done as given below.

• Step1: Calculate the duration on and off times of all units for the whole schedule time horizon.

• Step2: set h = 1

• Step3: set iteration count n = 1

• Step4: $if{u}_n^h=0{\displaystyle \begin{array}{c}\begin{array}{l}\\ {}\end{array}\end{array}}$and ${u}_n^{h-1}=1$ and ${T}_{n, on}^{h-1}\le MUT$ then set ${u}_n^h=1$

• Step5: if ${u}_n^h=0$ and ${u}_n^{h-1}=1$ and h + MDT − 1 ≤ T and ${T}_{n, off}^{off+ MDT-1}\le MDT$ SET ${u}_n^h=1$

• Step6: if $if{u}_n^h=0$ and ${u}_n^{h-1}=1$ and t + MDT − 1 > T and ${\sum}_{n=h}^H{u}_n^h>0$ set ${u}_n^h=1$

• Step7: update the time period of ON/OFF times for unit i

• Step8: Do n = n + 1 return to step 4.

• Step9: $ifh<H,{\displaystyle \begin{array}{c}h=h+1,\end{array}}$ return to step 3,

• Step10: If condition at step 9, found false, stop.

6 Test system and standard benchmark

In order to validate the performance of the proposed hGWO-RES algorithms, 23 benchmark functions [2] has been taken into consideration and has been shown in Tables 2, 3, and 4. Table 2 depicts the Unimodal Benchmark Function, Table 3 depicts the Multi-modal Benchmark functions and Table 4 depicts the fixed dimensions benchmark problems.

Table 2 Unimodal benchmark function

Table 3 Multi-modal benchmark functions

Table 4 Fixed dimension benchmark function

In order to show the efficacy of the anticipated hGWO-RES algorithms for generation scheduling and dispatch problem, different types of test systems have been taken into consideration, which includes 7-, 10-, 19-, 20- and 40-Generating Units system [58]. The load demand pattern of 24-h are taken into consideration for effective research study. In the whole research study, 30 search agents are taken into considerations and algorithm is simulated for maximum iterations of 500.

7 Results and discussion

In order to overcome the stochastic nature of proposed hGWO-RES algorithm and validate the results, 30 trial runs are taken into consideration and each objective function has been evaluated for average, standard deviation, worst and best values. In order to validate the exploitation phase of proposed algorithm, unimodal benchmark function F1, F2, F3, F4, F5, F6 and F7 are taken into consideration. Table 5 shows the solution of unimodal benchmark function using hGWO-RES algorithm. The comparison results for unimodal benchmark functions has been shown in Table 6, which are compared with other recently developed metaheuristics search algorithms GWO [2], PSO [55], BA [33], FPA [21], GA [85], DA [6], BDA [6], BPSO [86] and BGSA [39] in terms of average and standard deviation. The convergence curve and trial solutions for hGWO-RES for unimodal benchmark functions are shown in Figs. 5 and 6.

Table 5 Results of hybrid GWO-RES algorithm for unimodal benchmark function

Table 6 Comparison of unimodal benchmark functions

Fig. 5

Convergence curve of hGWO-RES for unimodal benchmark functions

Fig. 6

Trial solutions for unimodal benchmark functions

In order to validate the exploration phase of proposed algorithm, the multi-modal benchmark functions F8, F9, F10, F11, F12 and F13 are taken into consideration, as these functions have many local optima with the number increasing exponentially with dimension. Table 7 shows the solution of multi-modal benchmark function using hGWO-RES algorithm. The comparison results for multi-modal benchmark functions has been shown in Table 8, which are compared with other recently developed metaheuristics search algorithms GWO [2], PSO [55], BA [33], FPA [21], GA [85], DA [6], BDA [6], BPSO [86] and BGSA [39] in terms of average and standard deviation. The convergence curve and their corresponding trial solutions of hGWO-RES for multi-modal benchmark functions are shown in Figs. 7 and 8.

Table 7 Results of hybrid GWO-RES algorithm for multi modal benchmark function

Table 8 Comparison of multi modal benchmark functions

Fig. 7

Convergence curve of hGWO-RES for multi-modal benchmark functions

Fig. 8

Trial solutions of hGWO-RES for multi-modal benchmark functions

The test results for fixed dimension benchmark problems are shown in Table 9. The comparison results for multi-modal benchmark functions has been shown in Tables 10 and 11, which are compared with other recently developed metaheuristics search algorithms GWO [2], PSO [55], BA [33], FPA [21], GA [85], DA [6], FEP [87], GSA [35] and DE [52] in terms of average and standard deviation. The trial solutions for fixed dimension benchmark functions along with their convergence curve are shown in Figs. 9 and 10.

Table 9 Results of hybrid GWO-RES algorithm for fixed dimension benchmark function

Table 10 Comparison of fixed dimension benchmark functions

Table 11 Comparison of results for fixed dimension benchmark functions

Fig. 9

Convergence curve of hGWO-RES for fixed dimension benchmark functions

Fig. 10

Trial solutions of hGWO-RES for fixed dimension benchmark functions

In order to verify the performance of proposed hGWO-RES algorithm for generation scheduling and dispatch problems, the conventional UCP and UCP considering wind power as renewable energy sources are solved and their corresponding solutions are represented in Tables 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, and 28 . Table 29 depicts the optimal cost analysis for 10-generating unit systems considering wind power have been compared with recently developed algorithms.

Table 12 Results for 56-bus system using hGWO-RES (considering thermal units)

Table 13 Results for 56-bus system using hGWO-RES (considering wind-thermal units)

Table 14 Thermal commitment and generation schedule for 10-unit test system with 5% spinning reserve using hGWO-RES

Table 15 Wind-thermal commitment and generation schedule for 10-unit test system with 5% spinning reserve using hGWO-RES

Table 16 Thermal commitment and generation schedule for 10-unit test system with 10% spinning reserve using hGWO-RES

Table 17 Wind-thermal scheduling and dispatch for 10-unit test system with 10% spinning reserve using hGWO-RES

Table 18 Commitment status and generation schedule of IEEE-118 bus system using hGWO-RES (considering thermal units)

Table 19 Commitment status and generation schedule of IEEE-118 bus system using hGWO-RES (considering wind-thermal)

Table 20 Commitment status of 20 unit thermal unit with 10% spinning reserve using hGWO-RES

Table 21 Generation scheduling of 20 unit thermal unit with 10% spinning reserve using hGWO-RES

Table 22 Wind-thermal commitment status for 20-unit test system with 10% spinning reserve using hGWO-RES algorithm

Table 23 Wind-thermal generation schedule for 20-unit test system with 10% spinning reserve using hGWO-RES algorithm

Table 24 Commitment status of 40 unit with 10% spinning reserve using hGWO-RES (considering thermal units)

Table 25 Generation scheduling of 40 unit with 10% spinning reserve using hGWO-RES (considering thermal units)

Table 26 Generation scheduling of 40-unit with 10% spinning reserve using hGWO-RES (considering wind-thermal system)

Table 27 Generation scheduling of 40-unit with 10% spinning reserve using hGWO-RES (considering wind-thermal)

Table 28 Comparative cost analysis for various test systems (% saving in cost)

Table 29 Comparative cost analysis for 10-unit system considering wind power (for 10% spinning reserve)

8 Conclusion

In the proposed research, a hybrid version of grey wolf optimizer with random exploratory search has been presented to solve benchmark problems and generation scheduling and dispatch problem of electric power system with due consideration of wind energy as renewable energy source. Results of hGWO-RES has been tested for non-linear, highly constrained, non-convex engineering design and optimization problems, which include 23 benchmark problems and combinatorial unit commitment problem of electric power system. Experimentally, it has been found that the exploitation phase of the existing GWO algorithm has been improved. Also, the authors have presented the solution of scalar objective generation scheduling and dispatch of thermal generating units considering impact of wind power using hGWO-RES algorithm. The results for IEEE test system consisting of 7-, 10-, 19-, 20- and 40-generating units has been evaluated, analyzed for percentage cost saving and has been compared with recently developed algorithms, while considering wind power as a renewable energy source. Also, it is evident from analysis that by integrating wind power source along with conventional thermal power system, power production cost per megawatt is significantly reduced. Hence it is envisaged to incorporate wind energy source to tackle price hiking problem. Moreover, hGWO-RES accelerates the progress towards the near global optimum point thereby enabling one to obtain improved solutions with a reduced computation overhead.