Optimum generation scheduling incorporating wind energy using HHO–IGWO algorithm

Abstract

Recently, renewable energy participation is gaining importance in the existing power system. However, the large penetration of these renewable energy sources into the existing power system network may cause an imbalance in supply and demand response. Unit commitment is the decision-making process in which generating units are turned ON and OFF at the hourly interval as per the load demand under certain constraints to provide economic scheduling. Thus, an advanced intelligent approach is needed to cope with this combined unit commitment problem with a large penetration of intermittent sources. This paper offers the solution to optimal scheduling by implementing the hybrid Harris Hawks optimizer algorithm (HHO–IGWO). Standard IEEE systems with 10-, 19-, 20-, and 40 units are simulated. Further, to test the feasibility and effectiveness of the proposed method, a comparative analysis for a 10-, 20-, and 40-unit system has also been performed with penetration. The comparative analysis reveals that proposed is more efficient in tackling unit commitment problem in the presence of wind as renewable energy source.

Introduction

Electric power plays a vital role in the development, modernization, and progress of upgraded technology. Conventional energy sources such as coal, oil, and gas are rapidly exhausting. The systematic management of the generation schedule within the constraints comes under the category of unit commitment. Unit commitment problem is a complex problem where the generation schedule is planned well in advance with sufficient spinning reserve to satisfy sudden increase in demand [1]. Sustainable power sources such as wind and solar are gaining more significance as these sources are inexhaustible and provide economic operation. Wind energy is getting more attention in the power sector, as wind power helps to reduce the burden on conventional fuels and also decreases environmental pollutants. But, due to the stochastic nature of wind energy, constant power is not available at all times, which results in even more complexity. As the manual calculations require large amounts of computation time to solve unit commitment problems involving wind power, a computer-based system used to solve UC problems not only enhances the computational capability but also improves solution efficiency and reliability with a proper logical approach.

Optimization is the process in which a particular objective function is solved by applying a defined algorithm to get the optimal solution. Over a few decades, several heuristics and metaheuristics have been built by researchers to handle various optimization problems using globally accepted algorithms, such as binary bat algorithm [2], salp swarm algorithm [3], ant colony optimization [4], shuffled frog leaping algorithm [5], biogeography-based optimization [6], gravitational search algorithm [7], differential evolution algorithm [8], particle swarm optimization [9]. Baldwin et al. [10] were the first who published a paper in the field of unit commitment in the year 1959. Priority method, dynamic programming, Lagrange relaxation, and branch & bound methods are the foremost methods to solve the unit commitment problem. Afterward, new optimization methods such as genetic algorithm, simulated annealing, analytical hierarchy process, and particle swarm optimization were implemented by researchers to solve the unit commitment problem more precisely. Recently, due to the introduction of renewable energy sources such as wind and solar, the load demand on conventional sources has reduced to a large extend. Some technical findings related to thermal-wind commitment are discussed as follows:

Dieu et al.[11] have presented a primary generating schedule excluding start-up and shunt down constraints that is updated by IPL and ALH to resolve ramp rate constraint commitment. The hybrid algorithm is found to be effective in providing increased spinning reserve. Yuan et al. [12] proposed the IBPSO method in which unit characteristics are enhanced by using BPSO for tackling unit commitment problem and heuristic lambda-iteration method for economic load dispatch problem. Tan et al. presented a solution for optimal allocation and sizing of renewable DG sources in various distribution networks by the ant lion optimization algorithm (ALOA) [13]. Entezariharsini et al. elaborated impacts of increased wind power in terms of the level of penetration. Stochastic programming including wind power uncertainty is presented to minimize the annual operational cost of generators [14]. Bhadoria et al. utilized the inherent property of moths to converge toward the light to solve the economic load dispatch problem with due consideration of renewable energy sources [15]. Anand et al. have combined the exploration capability of particle swarm optimizer (PSO) and exploitation competency of sine–cosine algorithm (SCA) to form hybrid civilized swarm optimization algorithm. Reddy et al. [16] have presented sigmoid and tangent hyperbolic transfer functions and applied three binary gray wolf optimizer (BGWO) models to solve the profit-based self-scheduling problem of generation. Suresh et al. have modeled a hybrid system consisting of wind and solar by implementing probability distribution methods using diverse probability.

These optimization methods are found to be efficient in solving complicated generation scheduling issues. But, one of the major issues with these techniques is their inefficiency in finding local optimal points during the search process. Eventually, the No-Free-Lunch theorem permits the design of new algorithms as no single algorithm is efficient enough to solve all optimization issues. This motivates us to solve the combinational unit commitment problem using a hybrid variant of Harris Hawks optimizer (HHO) and an improved gray wolf optimizer (IGWO).This paper offers the solution to the unit commitment problem incorporating wind by using the proposed HHO–IGWO algorithm. Standard IEEE systems consisting of small, medium, and large systems, which include 10-, 19-, 20-, and 40 units, are simulated in MATLAB software using a hybrid HHO–IGWO algorithm with and without wind penetration. The unit commitment problem formulation with wind penetration is discussed in the subsequent section.

Construction of unit commitment problem

Unit commitment problem is an optimization problem in which, the generated power is systematically distributed for a forecasted load to minimize the overall cost of power generation while satisfying all equality and inequality constraints. The major objective of unit commitment problem is selecting a proper generating schedule to minimize the total power generation cost. The total fuel cost $F_{{\text{T}}}$ is determined using Eq. (1) by summing up the generation cost of each unit for a defined time interval [17].

$$F_{T} = \sum\limits_{h = 1}^{H} {\left( {\sum\limits_{i = 1}^{N} {\left[ {\left( {a_{i} \,P_{i,h}^{2} + \,b_{i} \,P_{i,h} + c_{i} } \right)U_{i,h} \, + \,{\text{STC}}_{i} (1 - U_{i(h - 1)} )\,U_{i,h} } \right]\$ {\text{/hr}}} } \right)}$$

(1)

where $a_{i} ,\,b_{i} \,{\text{and}}\,c_{i}$ are the fuel cost function expressed in $/h, $/MWh, and $/MWh² respectively.

Mathematically, start-up cost ${\text{STC}}_{i}$ [18] is expressed as the sum of Hot start-up cost $\left( {{\text{HSc}}_{i,h} } \right)$ and $\left( {{\text{CSc}}_{i,h} } \right)$ ith unit respectively.

$${\text{STC}}_{i} = \left\{ {\begin{array}{*{20}l} {{\text{HSc}}_{i,h} ;} \hfill & {{\text{MD}}t_{i} \, \le \,T_{i.h}^{{{\text{OFF}}}} \,\, \le \,\,(\,{\text{MD}}t_{i} + \,{\text{CSh}}_{i} )\quad (i = \,N;h = 1,2,3, \ldots ,H)} \hfill \\ {{\text{CSc}}_{i,h} ;} \hfill & {T_{i,h}^{{{\text{OFF}}}} \, > \,\left( {{\text{MDt}}_{i} + \,{\text{CSh}}_{i} } \right)} \hfill \\ \end{array} } \right.$$

(2)

The power balance is achieved when overall generation meets the allocated load as expressed in Eq. (3) [19],

$$\sum\limits_{i = 1}^{N} {{\text{PG}}_{i} } \, \cdot U_{i,h} + P_{g}^{w} = \,D_{L} \quad \left( {i = 1,2, \ldots ,N;\,h = 1,2,3....,H} \right)$$

(3)

For arbitrary free unit power outputs, within minimum and maximum power limit,

${\text{PG}}_{i}^{\min } \le \,{\text{PG}}\, \le \,{\text{PG}}_{i}^{\max } \,\,(i = 1,2, \ldots ,N\,\,\,;\,\,h = 1,2, \ldots ,H)\,\,$, it is assumed that the Rth reference unit power output is constrained by the power balance Eq. (4) [19].

$$P_{{{\text{hR}}}} = D_{L} - \sum\limits_{\begin{subarray}{l} i = 1 \\ i \ne R \end{subarray} }^{{{\text{NG}}}} {(P_{g(i)} U_{i,h} + P_{g}^{w} )} \quad (h = 1,2, \ldots ,H)$$

(4)

In order to mitigate unpredictable disturbances such as sudden load demand or unexpected tripping of lines or generators, some additional generation capacity must be readily available. This additional generation capacity is referred to as spinning reserve. Due to wind penetration, some additional power is accessible from this renewable energy source. This additional power contributed by wind energy results in reducing the liability on thermal units. Equation (6) signifies that the total available generation should always be equal to or greater than sum of load demand and spinning reserve [19].

$$\sum\limits_{i = 1}^{N} {P_{g\;\max (i)} } U_{i,h} + P_{g(h)}^{W} \ge D_{L(h)} + {\text{SR}}_{(h)} \quad \left( {h = 1,2, \ldots H} \right)$$

(5)

Generators cannot be turn-on and turn-off instantly. Minimum up time $\left( {{\text{MUT}}} \right)$ is the time to set a generating unit online after it has already been shut down [19].

$$T_{i,h}^{{{\text{ON}}}} \,\, \ge \,{\text{MUT}}$$

(6)

Similarly, the minimum down time $\left( {{\text{MDN}}} \right)$ is the amount of time for which a particular unit should be kept in off condition before putting it online [19].

$$T_{i,h}^{{{\text{OFF}}}} \,\, \ge \,{\text{MDN}}$$

(7)

Mathematical modeling of uncertainties of wind power

Wind power can be evaluated by probability distribution function which is mathematically represented as,

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

(8)

As the power generated by wind is an uncertain due to the randomness of wind speed, which is mathematically described as [15],

$$Pw = \left\{ {\begin{array}{*{20}c} 0 & {(v^{h} \, \le \,\,v_{{{\text{in}}}} \,\,or\,\,\,v^{h} \,\, \ge \,\,v_{{{\text{out}}}} )} \\ {P_{wr} } & {(v_{r} \, \le \,\,\,\,v^{h} \,\, \ge \,\,v_{{{\text{out}}}} )} \\ {\frac{{\left( {v - v_{{{\text{in}}}} } \right)}}{{v_{r} - v_{{{\text{in}}}} }}} & {(v_{{{\text{in}}}} \, \le \,\,\,\,v^{h} \,\, \ge \,\,v_{r} )} \\ \end{array} } \right.$$

(9)

From Eq. (9), when wind speed v^h is less than or equal to minimum rated velocity, wind power is zero. The probability of wind power being 0, p_wr be calculated as per Eqs. (10) and (11) respectively [20].

$$P_{r} (P_{w} = 0)\, = \,{\text{cdf}}(v_{{{\text{in}}}} )\, + \left[ {1 - {\text{cdf}}(v_{{{\text{out}}}} )} \right]$$

(10)

$${\text{For}}\,P_{w} \, = 0,\Pr \,\left[ {1 - \exp \left[ { - \left( {\frac{{v_{{{\text{in}}}} }}{\lambda }} \right)^{k} } \right.} \right]\,\, + \,\exp \left[ { - \left( {\frac{{v_{{{\text{out}}}} }}{\lambda }} \right)^{k} } \right]$$

(11)

The probability density function $\left( {{\text{pdf}}} \right)$ in Eq. (12) [20] depends upon $v_{{{\text{in}}\,\,}} {\text{and}}\,v_{r}$ due to randomness in wind speed.

$${\text{pdf}}(P_{W} )\, = \,\frac{{{\text{KL}}v_{in} }}{{\left( {P_{{{\text{wr}}}} } \right)^{\lambda } }}\,\left[ {\frac{{1 + \left( {{\raise0.7ex\hbox{${{\text{LP}}_{w} }$} \!\mathord{\left/ {\vphantom {{{\text{LP}}_{w} } {P_{{{\text{WR}}}} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${P_{{{\text{WR}}}} }$}}} \right)v_{{{\text{in}}}} }}{\lambda }} \right]\, \times \,\exp \left[ { - \left( {\frac{{1 + \left( {{\raise0.7ex\hbox{${{\text{LP}}_{W} }$} \!\mathord{\left/ {\vphantom {{{\text{LP}}_{W} } {P_{{{\text{WR}}}} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${P_{{{\text{WR}}}} }$}}} \right)\,v_{{{\text{in}}}} }}{\lambda }} \right)^{k} } \right]$$

(12)

Since output power delivered by wind generator is never remains constant and continuously fluctuates over an entire period, exact wind power extrapolation is not possible. The subsequent section presents mathematical formulation of HHO–IGWO.

Mathematical formulation of hybrid HHO–IGWO algorithm

HHO has inherent proficiency of proper balance between intensification and diversification. Studies revealed that slow convergence gives rise to reduced computational efficiency. The HHO algorithm does not need initial values for the judgment variables and exploit a stochastic indiscriminate search instead of using gradient search [21]. In Eq. (13a), when $q\, \ge \,0.5$ or perch on randomly on tall trees and modeled as in Eq. (13b) for $q\, < \,0.5$ [22].

$$X({\text{itn}} + 1) = \left\{ {X_{{{\text{rand}}}} } \right.({\text{itn}}) - r_{1} \times {\text{abs}}(X_{{{\text{rand}}}} ({\text{itn}}) - 2 \times r_{2} \times X({\text{itn}}))\begin{array}{*{20}c} ; & {q \ge 0.5} \\ \end{array}$$

(13a)

$$X({\text{itn}} + 1) = \left\{ {(X_{{{\text{prey}}}} } \right.({\text{itn}}) - X_{m} ({\text{itn}})) - r_{3} \times (L{\text{b}} + r_{4} \times ({\text{Ub}} - {\text{Lb}}));\quad q < 0.5$$

(13b)

where $X({\text{itn}} + 1)$ is the Hawks position in ensuing iteration $\left( {{\text{itn}}} \right)$, $X_{{{\text{rand}}}} ({\text{itn}})$ is randomly selected Hawks, corresponding to the vectors $r_{1} ,\,r_{2} ,\,r_{3} ,\,r_{4} ,\,{\text{and}}\,q$ are random values in between (0, 1) and these are modified in each iteration between upper bound $\left( {{\text{Ub}}} \right)$ and lower bound $\left( {{\text{Lb}}} \right)$. $X_{{{\text{prey}}}} \left( {{\text{itn}}} \right)$ denotes the position of prey.$X_{m} \left( {{\text{itn}}} \right)$ epitomizes the mean position of Hawks which is determined using Eq. (14) [23].

$$X_{m} ({\text{itn}}) = \frac{1}{N}\left( {\sum\limits_{i = 1}^{N} {X_{i} } ({\text{itn}})} \right)$$

(14)

Changeover from exploration to exploitation phase depends upon the fugitive energy of the target, assessed using Eqn. (15) [23].

$$E_{A} = 2 \times E{}_{0} \times \left( {1 - \frac{{{\text{itn}}}}{{{\text{itn}}_{\max } }}} \right)$$

(15)

where $E_{A}$ is evading energy of the prey, $E_{0}$ is the initial energy of the prey changing randomly between (− 1, 1) and ${\text{itn}}_{\max }$ is maximum iterations. Equation (16) is used to determine the upgraded position of Hawks. The successful capture relies on attacking strategies of Hawks and escaping nature of prey depending upon change of escape (r). Hawks perform a soft besiege for $r \ge 0.5\& \left| E \right| \ge 0.5$ [24].

$$X({\text{itn}} + 1) = \Delta X({\text{itn}}) - E_{A} \times {\text{abs}}(J\, \times X_{{{\text{prey}}}} ({\text{itn}}) - X({\text{itn}}))$$

(16)

$$\Delta X({\text{itn}}) = \left( {X_{{{\text{prey}}}} ({\text{itn}}) - X({\text{itn}})} \right)$$

(17)

where $\Delta X({\text{itn}})$ is the variance between current location of prey and locality of Hawks at iteration ${\text{itn}}$. J = 2(1 − r) is the Jump energy which modifies randomly in every iteration. $r_{5}$ is the random numeral in the range (0, 1). The tired target fails to escape and Hawks perform hard besiege as modeled in Eq. (18). Hawks perform a hard besiege for $r \ge 0.5\& \left| {\,E\,} \right| < 0.5$ [24].

$$X(itn + 1) = X_{{{\text{prey}}}} (itn) - E_{A} \times abs(\Delta X(itn))$$

(18)

$$Y = X_{{{\text{prey}}}} ({\text{itn}}) - E \times {\text{abs}}(JX_{{{\text{prey}}}} ({\text{itn}}) - X({\text{itn}}))$$

(19)

$$Z = Y + S \times L_{F} (D)$$

(20)

where $Y$ and $Z$ are the positions based on soft besiege.

The $L_{F} \left( D \right)$-based designs which follow the certain rule [25]. At this stage, the prey has enough energy and besiege during this phase depends on levy flight (LF) concept as modeled in Eq. (21) [25]. Hawks perform a soft besiege through rapid dives for $\left| {\,E\,} \right| \ge 0.5\,\,\& r < 0.5$.

$$X({\text{itn}} + 1) = \left\{ \begin{gathered} Y\begin{array}{*{20}c} ; & {{\text{if}}} & {F(Y) < F(X({\text{itn}}))} \\ \end{array} \hfill \\ Z\begin{array}{*{20}c} ; & {{\text{if}}} & {F(Z) < F(X({\text{itn}}))} \\ \end{array} \hfill \\ \end{gathered} \right.$$

(21)

$$Y = X_{{{\text{prey}}}} ({\text{itn}}) - E \times {\text{abs}}(JX_{{{\text{prey}}}} ({\text{itn}}) - X_{m} ({\text{itn}}))$$

(22)

$$Z = Y + S \times L_{F} (D)$$

(23)

where $Y$ and $Z$ are the positions based on hard besiege.

The Hawks are very close to prey and perform hard besiege as modeled in Eq. (24). Hawks perform hard besiege through rapid dives for $\left| {E\,} \right| < 0.5\,\,\& r < 0.5$.

$$X({\text{itn}} + 1) = \left\{ \begin{gathered} Y^{\prime } \begin{array}{*{20}c} ; & {{\text{if}}} & {F(Y^{\prime } ) < F(X({\text{itn}}))} \\ \end{array} \hfill \\ Z^{\prime } \begin{array}{*{20}c} ; & {{\text{if}}} & {F(Z^{\prime } ) < F(X({\text{itn}}))} \\ \end{array} \hfill \\ \end{gathered} \right.$$

(24)

where $Y^{\prime }$ and $Z^{\prime }$ are the positions based on hard besiege.

Updating $X\left( {{\text{iter}} + 1} \right)$ by improved gray wolf optimizer (IGWO)

At this stage, a weighted average of alpha, beta, and delta wolfs is evaluated and then best individual is assigned a weight, obtained by multiplying its corresponding vectors ‘A’ and ‘C’. The best fitness value of gray wolves depends upon the fitness value evaluated as ‘a’ shown in eqn. (25). Mathematically, $(\overrightarrow {{G_{w} }} )\& \overrightarrow {W}_{{G({\text{itn}} + 1)}}$ vectors are defined through Eqn. (27) to (28) [25].

$$a = 2 - t \times \left( {\frac{2}{{{\text{itn}}_{max} }}} \right)$$

(25)

$$\overrightarrow {{G_{W} }} \,\, = \left| {C \times W_{{{\text{prey}}}} \left( {{\text{itn}}} \right) - W_{G} \left( {{\text{itn}}} \right)} \right|$$

(26)

$$W_{G} \left( {{\text{itn}} + 1} \right) = \,W_{{{\text{Prey}}}} \left( {{\text{itn}}} \right) - \overrightarrow {A} \times G_{W}$$

(27)

The extreme search process takes place and various fitness values for $(\overrightarrow {{W_{\alpha } }} )$, $(\overrightarrow {{W_{\beta } }} )$ and $(\overrightarrow {{W_{\delta } }} )$ are updated using Eqs. (31), (33) and (35).The final position for capturing the prey is evaluated by Eq. (36).

$$G_{\alpha } = {\text{abs}}\left( {\overrightarrow {{C_{1} }} \cdot \overrightarrow {W}_{\alpha } - \overrightarrow {{W_{G} }} } \right)$$

(28)

$$\overrightarrow {{W_{1} }} \, = \,\overrightarrow {{W_{\alpha } }} - \overrightarrow {{A_{1} }} \cdot \overrightarrow {{G_{\alpha } }}$$

(29)

$$G_{\beta } = {\text{abs}}\left( {\overrightarrow {{C_{2} }} \cdot \overrightarrow {W}_{\beta } - \overrightarrow {{W_{G} }} } \right)$$

(30)

$$\overrightarrow {{W_{2} }} \, = \,\overrightarrow {{W_{\beta } }} - \overrightarrow {{A_{1} }} \cdot \overrightarrow {{G_{\beta } }}$$

(31)

$$G_{\delta } = {\text{abs}}\left( {\overrightarrow {{C_{3} }} \cdot \overrightarrow {W}_{\delta } - \overrightarrow {{W_{G} }} } \right)$$

(32)

$$\overrightarrow {{W_{3} }} = \,\,\overrightarrow {{W_{\delta } }} - \overrightarrow {{A_{3} }} \cdot \overrightarrow {{G_{\delta } }}$$

(33)

$$\overrightarrow {W}_{{\left( {itn} \right)}} = \left( {\frac{{\overrightarrow {W}_{1} + \overrightarrow {W}_{2} + \overrightarrow {{W_{3} }} }}{3}} \right)$$

(34)

The pseudocode of the proposed metaheuristic algorithm has been depicted in the flowchart as shown in Fig. 1. In this study, sequential hybridization is utilized for initializing gray wolf position, and thereafter, updated gray wolf positions are more intensively explored by the search agents.

Fig. 1

Flowchart for HHO–IGWO algorithm

Implementation of proposed HHO–IGWO algorithm for unit commitment problem

The HHO–IGWO method is a metaheuristic algorithm that has an excellent ability of exploration and exploitation and effectively utilized to solve the unit commitment problem [26].

Pseudocode to repair spinning reserve, MDT, MUT, constraints

Once a unit is started, it should not be turned off immediately before reaching MUT. This is required to satisfy economic, mechanical, and design limitations. Similarly, any unit which is once de-committed should not put online immediately. The HHO–IGWO algorithm may sometimes perform unenviably to satisfy the spinning reserve constraint. However, some repair in minimum up/down constraint, excessive spinning reserve is needed. The flowchart for spinning reserve repairing is illustrated in Fig. 2.

Fig. 2

Flowchart for spinning reserve repairing

De-committing of excess of units

During the repair process of MDT/MUT and spinning reserve, some of the units may get unnecessarily ON. To avoid this situation that could result in an excessive cost for running those units, some of the units need to be shut down. Figure 3 shows the flowchart for de-committing excessive spinning reserve. Figure 4 shows the flowchart of entire process of commitment using HHO–IGWO.

Fig. 3

Flowchart for the de-commitment of excessive generating units

Fig. 4

Flowchart of entire process of commitment using HHO–IGWO

Results and discussion

In this section, the test results of standard IEEE with 10, 19, 20 and 40 thermal units along with wind penetration are analyzed. The test systems are simulated in MATLAB 2018a Windows 10, CPU@2.10Ghz-4 GB RAM Core i5. To check the performance of the HHO–IGWO method for solving the optimal scheduling, the standard test system of IEEE is taken into consideration. Table 1 illustrates generation scheduling for 10 generating units, and Table 2 shows generation scheduling for 10 generating units with wind penetration. From Tables 1 and 2, it can be seen that the cost of generation with thermal units is 563435.9964 $ per hour, and the cost of generation for the same number of units with wind penetration is 492400.2699 $ per hour. This suggests that there is a cost-saving of 71,035.7265($/hr) and for 8760 h per year the total saving in cost is 622272964.14($/year).

Table 1 Power scheduling for 10-unit test system with (10% SR) using hHHO–IGWO

Table 2 Power scheduling for 10-unit system with wind penetration using hHHO–IGWO

Table 3 illustrates generation scheduling for 19 generating units with 10% SR for thermal-wind system. The cost of generation with thermal units is 207001.8242 $ per hour, and the cost of generation for the same number of units with wind penetration is 196723.619 $ per hour. This suggests that there is a cost-saving of 10,278.2052($/hr) and for 8760 h per year the total saving in cost is 90037077.552($/year).

Table 3 Power scheduling of 19-unit system with wind penetration using hHHO–IGWO

Table 4 illustrates generation schedule for 20 generating units with wind penetration. The cost of generation with thermal units is 1127513.692$ per hour, and the cost of generation for the same number of units with wind penetration is 1052906.5262 $ per hour. This suggests that there is a cost-saving of 71,954.1638($/hr) and for 8760 h per year the total saving in cost is 630318474.888($/year).

Table 4 Power scheduling of 20-unit system with wind penetration using hHHO–IGWO

In Tables 5 and 6 illustrates generation schedule for 40 generating units with wind penetration. The cost of generation with thermal units is 2249657.3623 $ per hour. From Table 6, it can be seen that the cost of generation for the same number of units with wind penetration is 2172361.1608 $ per hour. This suggests that there is a cost-saving of 77,296.2015($/hr) and for 8760 h per year the total saving in cost is 67714725.14($/year).

Table 5 Power scheduling for 1–20 units with wind using hHHO–IGWO

Table 6 Power scheduling for 21–40 units with wind using hHHO–IGWO

Table 7 illustrates percentage cost-saving for 10-, 19-, 20-, and 40- units with wind using hHHO–IGWO. It shows with 10-unit system, there is % cost-saving of 12%. For 19 units, there is a % cost-saving of 4.90%. For 20 units, there is a % cost-saving of 6.5% while in case of 40 units, 3.2% cost-saving is noted. It is observed that proposed algorithm is efficient in solving unit commitment problem with more precision and accuracy.

Table 7 Percentage cost-saving for 10-, 19-, 20-, and 40 units using hHHO–IGWO

Table 8 shows a cost comparison of 10 units (10% SR) for power generation with wind penetration. In Table 8, best, worst, and mean values for various methods are presented. Results illustrated in Table 8 reveal that the proposed method is more effective in solving unit commitment problem as compared to other known techniques.

Table 8 Comparison of 10-unit wind–thermal system with other algorithms

Table 9 shows a cost comparison of 20 units (10% SR) for power generation with wind penetration. In Table 9, best, worst, and mean values for various methods are presented. Results illustrated in Table 8 show that HHO–IGWO gives total generation cost wind penetration as 1,052,906.52$ which is less than BMFO-SIG, hGWO-RES, and GWO methods. The comparative analysis reveals that proposed method is efficient in resolving unit commitment problem with large wind penetration.

Table 9 Comparison of 20-unit wind–thermal system with other algorithms

Similarly, Table 10 illustrates a cost comparison of 40 units (10% SR) for power generation with wind penetration. Results illustrated in Table 8 show that HHO–IGWO gives total generation cost wind penetration as 1,052,906.52$ which is less than BMFO-SIG and hGWO-RES methods. This suggests that that the proposed method is more effective in solving unit commitment problem when compared to other competent methods.

Table 10 Comparison of 40-unit wind–thermal system with other algorithms

Conclusion

In this research work, a novel hybrid optimization technique based on the integration of HHO and IGWO has been utilized effectively to solve the UC problem. Four standard IEEE test systems are simulated with due effect of wind power penetration into the existing conventional thermal system consisting of 10-, 19-, 20-, and 40 units. The analysis shows that the proposed hybrid metaheuristic algorithm is efficient to provide a cost effective solution for handling the unit commitment problem. Further, to investigate the validity of the proposed algorithm, a comparative analysis for a 10-, 20-, and 40-unit system with wind penetration is also been performed. The comparative study reveals that the proposed algorithm is a promising technique to solve the UC problem with renewable energy penetration.