A binary bat algorithm with improved crossover operators and Cauchy mutation for unit commitment problem

Abstract

Power system operators are faced with the problem of unit commitment belonging to mixed integer programming, which becomes very complicated, as units become large-scale and highly constrained. Because unit commitment problem is a binary problem with commitment and de-commitment, a discrete/binary optimization algorithm with superior performance is required. This paper proposes a novel hybrid binary bat algorithm for unit commitment problem, which consists of two process. To begin with, the proposed binary bat algorithm is applied to determining the commitment schedule of unit commitment problem. Specifically, an improved crossover operator based on exponential-logic-modulo map is proposed to enhance the convergence and maintain the diversity of populations. To prevent the algorithm from falling into a local optimum, a local mutation strategy performs local perturbation. Chaotic map is responsible for updating some parameters to increase the performance of the proposed algorithm. Furthermore, Lambda-iteration method is adopted to solve economic load dispatch in continuous space. Constraint handling is performed using the heuristic constraint produce. The effectiveness of the proposed algorithm is verified by benchmark functions and test systems. Additionally, the simulation results are compared with other well-established heuristic and binary approaches.

Introduction

Power system is one of the indispensable infrastructures in modern society, and its stability and reliability are vital to the normal operation and development of society. Unit commitment (UC) problem is an important problem in power system, which is a complex and mixed integer programming optimization problem involving two optimization decision process, called the unit scheduling problem and economic load dispatch problem. It is a cost minimization problem that determines the start-up and shut-down schedule of generating units in order to minimize cost while meeting various system and unit operation constraints [1]. The constraints include load balance constraint, spinning reserve constraint, power generation limit, minimum up/down time constraint, and ramp up/down rate limit, etc. The operation and management of the power system benefit from an optimal UC solution in terms of cost savings and increased reliability. However, the better solution quality of UC problem is greatly affected by system dimension. That is not an easy task due to the large size of the problem and computational limitations. As a result, UC problem is known as one of the most difficult problems to solve in power systems.

There are many methods that have been developed to solve UC problem. The major methods include classical optimization methods like mixed integer linear programming (MILP) [2], priority list approach (PLA) [3], dynamic programming (DP) [4], branch-and-bound methods (BBM) [5] and Langrangian relaxation (LR) [6]. The advantages of classical methods lie in their simple application form and fast convergence. However, there are some drawbacks with poor quality solution (like PLA), “curse of dimensionality” (like DP), long execution time (like BBM), and approximation for handling nonlinear characteristics (like MILP). Specifically, the UC problem involves a 0–1 unit scheduling problem and an economic dispatch problem. When a large-scale UC problem need to be solved, numerical optimization methods alone may face “curse of dimensionality”, take much time to solve or even fail to obtain an optimal solution. Therefore, these methods alone may no longer be applicable for a large-scale UC problem.

Aside from the classical methods, heuristic approaches have grown significantly in recent years. Some of the heuristic approaches include evolutionary programming (EP) [7], genetic algorithm (GA) [8], simulated annealing algorithm (SA) [9], imperialistic competition algorithm (ICA) [10], gravitational search algorithm (GSA) [11, 12], ant colony search algorithm (ACSA) [13], and coyote optimization algorithm (COA) [14], etc. However, heuristic algorithms face the risk of local optimal and slow convergence. For example, GSA was introduced and applied successfully to find the minimum factor of the safety in slope stability analysis [11], but it has the risk of local optimal when the algorithm is applied to solve complex nonlinear mixed integer programming problems. Subsequently, hybrid algorithms are developed by combining the advantages of classical and heuristic algorithms. Specifically, a hybrid adaptive gravitational search algorithm integrating an adaptive gravitational search algorithm with pattern search was proposed to perform a multi-objective optimization of retaining walls successfully [12]. After experiments with five benchmark functions and two experiment cases, compared with other algorithms, the results show the superior performance of this hybrid heuristic algorithm. An improved manta ray foraging optimization (IMRFO) algorithm was designed to address the cost problem of power system congestion in [15]. It was experimentally verified that the method effectively reduced the congestion cost and possessed better convergence effect compared with other optimization methods. Considering the power system risk, a hybrid modified grey wolf optimization-sine cosine algorithm was presented to perform a multi-objective optimization of cost and risk in an integrated energy system [16]. The standard IEEE-30 bus and Indian-75 bus system were employed to validate the effectiveness of HMGWO-SCA. In Paul and Hati [17], authors proposed a hybrid Harris hawk optimization-sine cosine algorithms to perform the optimal scheduling of home energy management to reduce energy usage. In Paul et al. [18], a bus sensitivity factor and wind availability factor were integrated into whale optimization algorithm for achieving the optimal power cost rescheduling based on wind farm. In all the above studies, the improvement of hybrid heuristic algorithms are proposed successfully for various problems. However, it is necessary to propose an approach that is appropriate and efficient for UC problem.

There have been many heuristic algorithms applied to UC problem so far. Since UC is a mixed integer programming problem, researchers usually divide this problem into two subproblems, i.e. the unit scheduling problem and the economic dispatch problem. For example, in [19], the authors proposed a priority lists evolutionary algorithm (EA) for dealing with UC problem, in which priority lists were applied to steer the search towards adequate generating schedules. Besides, an elitist mutation strategy was proposed to boost the performance of the proposed EA-based approach. In Dhaliwal and Dhillon [20], to find the best profitable solution in profit-based UC problem, a new hybrid binary algorithm based on novel binary differential evolution (BDE) algorithm and an improved binary hill-climbing (BHC) algorithm was proposed. In Pan et al. [21], authors presented a new binary optimization algorithm named BFMO in terms of fish migration optimization algorithm. In addition, to prevent local traps of BFMO, long mutation and short mutation were adopted to further enhance the performance of BFMO. In Srikanth et al. [22], the author put forward quantum computing into a binary grey wolf optimizer for UC problem. Although the various binary optimization algorithms in the above studies are successfully applied to the UC problem, a binary algorithm with superior performance is especially important since in both subproblems. In particular, unit scheduling problem determines whether some units are started or not, thus affecting the optimal solution of UC problem.

Notably, a meta-heuristic algorithm named bat algorithm (BA) [23] was proposed to imitate the predatory behaviour of bats. The BA and its variants were applied to many practical engineering applications, such as economic dispatch [24, 25–26], structural optimization [27], unit commitment [28], and congestion alleviation [29]. Additionally, binary bat algorithm (BBA) is also developed in solving feature selection [30] and faulty identification [31]. However, the binary algorithm suffers from the same problem as the general continuous algorithm, i.e. it tends to fall into local optimums and cannot even obtain a globally optimal solution for increasingly complex optimization problems. Motivated by the characteristics and drawbacks of BBA, this paper presents a novel hybrid binary bat algorithm named NHBBA for unit scheduling problem. However, dimensions of economic dispatch problem are not complex as in the case of unit scheduling problem, which gives the possibility to use numerical optimization methods. Meanwhile, Lambda-iteration method has fast convergence and high-quality candidate solutions compared to other numerical optimization algorithms [32]. Therefore, Lambda-iteration method is employed to perform economic dispatch problem in this paper. Therefore, the main motivations are listed as follows:

1. The UC problem is known as a mixed integer programming problem in the power system, and it is also significant for the economy and reliability of the power system. However, the UC problem is vulnerable to “curse of dimensionality” using numerical methods, which can be time-consuming or even impossible to obtain an optimal solution. Besides, heuristic algorithms also face the risk of local optimal.

2. Binary bat algorithms have been used with excellent results in engineering applications, but they also face the challenge of local optimal when the dimensionality of the problem is too large.

According to the above motivations, the main contributions are listed as follows:

1. For the first time ever, NHBBA has been proposed and implemented to solve the UC problem.

2. To enhance convergence and maintain population diversity, a new crossover operator based on exponential-logic-modulo map is proposed.

3. To prevent local traps, Cauchy mutations are carried out.

4. The performance of NHBBA is enhanced by using a chaotic map to replace certain variables.

The rest of the paper is organized as follows. Section 2 describes the UC problem formulation and system and operation constraints. Section 3 shows the original bat algorithm and binary bat algorithm. Section 4 describes the proposed binary bat algorithm and implements UC problem by the proposed algorithm. Section 5 gives the simulation results for benchmark functions and different test systems. Section 6 concludes the paper.

Problem formulation

The UC problem is a power system scheduling problem that minimizes the start-up cost and fuel cost while satisfying system and operating constraints. The following section describes the objective and related constraints.

Objective function

The objective of UC problem is to find a minimum cost with fuel cost and start-up cost components.

Fuel cost function

By economically dispatching the units, the overall fuel cost for all committed generating units are minimized. The fuel cost function of unit i at hour t can be expressed as follows:

1

$$\begin{array}{c}\hfill C_{\text{f}}\left(P_i^t\right)=a_i+b_i\left(P_i^t\right)+c_i{\left(P_i^t\right)}^2\end{array}$$where $C_{\text{f}}$ is the total fuel cost, $a_i,b_i,c_i$ are the cost coefficients of thermal units, and $P_i^t$ is the active output of unit i at time t.

Start-up cost

In this paper, shut-down cost is neglected. Start-up cost is expressed as the cost of restarting de-commitment unit, which is associated with the number of hours. The start-up cost is described as follows:

2

$$\begin{array}{c}\hfill C_{\text{q}}^t=\left\{\begin{array}{c}{C}_{\text{q}}^{\text{hot}};T_{\text{down},i}^{\text{min}}\le T_{\text{off},i}\le \left\{C_{\text{q,i}}^{\text{cold}},+,T_{\text{down},i}^{\text{min}}\right\}\hfill \\ C_{\text{q}}^{\text{cold}};\left\{C_{\text{q,i}}^{\text{cold}},+,T_{\text{down},i}^{\text{min}}\right\}\le T_{\text{off},i}\hfill \end{array}\right)\end{array}$$where $C_{\text{q}}^{\text{hot}}$ denotes the hot start condition of the generation unit, $C_{\text{q}}^{\text{cold}}$ is the cold start condition of the generation unit, $T_{\text{off},i}$ is the generation unit’s de-committed time, $T_{\text{down},i}^{\text{min}}$ is the minimum time between two consecutive commitment generation units, and $C_{\mathrm{q},\mathrm{i}}^{\text{cold}}$ is the cold start hours of unit i.

Thus, the objective of minimizing the total cost is given by,

3

$$\begin{array}{c}\hfill min\text{TC}=\sum _{t=1}^T\sum _{i=1}^N{C}_{\text{f}}\left(P_i^t\right)S_i^t+C_{\text{q}}^t{S}_i^t\end{array}$$where t is the scheduling time index, T is total scheduling time, i denotes the index for generation unit, N is the number of generators, and $S_i^t$ is the on/off status of i at time t.

Constraints

Load constraint

Generator power from all committed units must meet the hourly load.

4

$$\begin{array}{c}\hfill \sum _{i=1}^N{P}_i^t{S}_i^t=P_{\text{sys}}^t\forall t\in T\end{array}$$where $P_{\text{sys}}^t$ is the system load demand at time t.

Spinning reserve constraint

It is necessary to maintain spinning reserves in a power system due to occasional equipment outages.

5

$$\begin{array}{c}\hfill \sum _{i=1}^N{S}_i^t{P}_{imax}\ge P_{\mathrm{sys}}^t+P_{\text{R}}^t\end{array}$$where $P_{\text{R}}^t$ is the spinning reserve at time t in UC problem.

Generation power limit

Each generation unit in committed has a generation limits, which is given by,

6

$$\begin{array}{c}\hfill S_i^t{P}_{min}^i\le P_t^i\le S_i^t{P}_{max}^i\end{array}$$where $P_{min}^i$ and $P_{max}^i$ are minimum and maximum generation limit of unit i, respectively.

Minimum up/down time constraint

It is predetermined how long should elapse between commitment and de-commitment events for a particular unit in order to ensure that it is reliable and performing satisfactorily.

7

$$\begin{array}{c}\hfill \begin{array}{c}\hfill S_i^{t+1}{T}_{\text{down},i}^{min}\le T_{\text{off},g}\text{if}{S}_i^t=0\\ \hfill \left(1,-,S_i^{t+1}\right)T_{\text{up},i}^{min}\le T_{\text{off},i}\text{if}{S}_i^t=1\end{array}\end{array}$$where $S_i^{t+1}$ is the status of unit i at time $t+1$, $T_{\text{down},i}^{min}$ and $T_{\text{up},i}^{min}$ are the minimum down time and minimum up time of unit i, respectively.

Overview of bat algorithm

This section introduces bat algorithm (BA) and the binary bat algorithm (BBA).

Original bat algorithm

Bat algorithm (BA) is a meta-heuristic algorithm that mimics the flight and hunting characteristics of bats. Based on the ultrasonic feedback to determine the location of the prey, bats then adjust these velocity and position. The frequency $F_k$, velocity $v_k^t$ and position $x_k^t$ of the BA mathematical model are defined as follows:

8

$$\begin{array}{cc}\hfill F_k=& F_{min}+\left(F_{max},-,F_{min}\right)\sigma ,\hfill \end{array}$$

9

$$\begin{array}{cc}\hfill v_k^{t+1}=& v_k^t+\left(x_k^t,-,G_{\mathrm{best}}^t\right)F_k,\hfill \end{array}$$

10

$$\begin{array}{cc}\hfill x_k^{t+1}=& x_k^t+v_k^{t+1}\hfill \end{array}$$where $F_{min}$ and $F_{max}$ are the minimum and maximum frequency, respectively, $\sigma$ is a stochastic values subject to [0,1], $G_{\mathrm{best}}^t$ denotes the best solution at time t in the population. Besides, a local search is executed to improve the exploration by random walk:

11

$$\begin{array}{c}\hfill x_{\text{new}}^{t+1}=G_{\mathrm{best}}^t+\xi A^t,\end{array}$$where $\xi$ is a randomly distributed value between 0 and 1, and $A^t$ is the average loudness of bat at time t. When the prey is found, the parameter A named loudness is maximum during the initialization and gradually decrease as the bat approaches prey. Meanwhile, the pulse emission rate r is increased. The two parameters are described as follows:

12

$$\begin{array}{cc}\hfill A_k^{t+1}=& \alpha A_k^t,\hfill \end{array}$$

13

$$\begin{array}{cc}\hfill r_k^{t+1}=& r_k^0(1-exp(-\theta t)),\hfill \end{array}$$where $\alpha$ belongs to [0,1], $\theta >0$, and $A_k^t\to 0$, $r_k^t\to r^0$ as $t\to \infty$.

Original binary bat algorithm

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Fig. 1

V-shaped transfer function in BBA

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Algorithm 1

Original binary bat algorithm

BA is a meta-heuristic algorithm for continue optimization problem. However, there are only two value, i.e. 0 and 1 in binary/discrete space which the condition must be addressed by the algorithm. As a result, the position updating rule (10) no longer works in binary/discrete space. This updating should be made in binary algorithms by means of individual velocities. The key is how to convert continuous variables (i.e. velocity vectors) into position variables (i.e. 0 or 1) in binary/discrete space. Transfer function is a common used way to deal with this issue [33]. In BBA [34], a v-shaped transfer function is proposed because it is a better way when velocity vector is with higher value (see Fig. 1). The v-shaped transfer function is shown as follows:

14

$$\begin{array}{cc}\hfill & V(v_k^l(t))=\left|\frac{2}{\pi },,\ast ,arctan,(\frac{\pi }{2}\ast v_k(t))\right|,\hfill \end{array}$$

15

$$\begin{array}{cc}\hfill & x_k^l(t+1)=\left\{\begin{array}{c}{x}_k^l{(t)}^{-1},\mathrm{if}\text{rand}<V(v_k^l(t+1)),\hfill \\ x_k^l(t),\mathrm{if}\text{rand}\ge V(v_k^l(t+1)),\hfill \end{array}\right)\hfill \end{array}$$where $x_k^l(t)$ and $v_k^l(t)$ denote the position and velocity of k-th bat at iteration t in the l-th dimension; $x_k^l{(t)}^{-1}$ indicates the complement of $x_k^l(t)$, and $\text{rand}$ is a random number in (0,1). The pseudo-code of BBA is shown in Algorithm 1.

[See PDF for image]

Algorithm 2

The proposed binary bat algorithm (NHBBA)

The proposed binary algorithm

An improved binary bat algorithm named NHBBA is proposed in this section. Specifically, the following steps are improved compared with original BBA: To enhance exploration, exploitation and convergence speed of NHBBA, an improved crossover operator based on exponential-logic-modulo map is proposed. BBA has the pros and cons of the continuous algorithm, so Cauchy mutations are performed to prevent local traps. Chaotic map is applied to improve the performance of NHBBA.

An improved crossover operator (NHBBA1)

One of the most important operators in the genetic algorithm is the crossover operator, which creates a new individual by exchanging the position information of the parents and carries on the parent’s valid information to the new individual [35]. The crossover operator-based hybrid meta-heuristic algorithm not only increases the diversity of the population, but also speeds up the convergence of the algorithm [36]. This paper proposes an improved crossover operator to update the velocity vector. The operator produces the offspring as follows:

16

$$\begin{array}{c}\hfill v_k^{t+1}=\left\{\begin{array}{c}C\ast v_k^t+(1-C)\ast G_{\mathit{best}}^t,\mathrm{if}\text{rand}<1/2,\hfill \\ C\ast G_{\mathrm{best}}^t+(1-C)\ast v_k^t,\mathrm{if}\text{rand}\ge 1/2,\hfill \end{array}\right)\end{array}$$where $v_k^t$ is the velocity of k-th bat at iteration t, $G_{\mathrm{best}}^t$ denotes the current group best, $r_k^t$ represents the pulse rates, and C belongs to [0,1].

According to Eq. (16), parameter C has a greatly influence in new individual, which determines the valid information given by the parents. To enable offspring to acquire more genes from superior parents and to maintain population diversity, this paper improves the original random parameters C. Exponential-logic-modulo map [37] is introduced to control the parameter C. The parameter C is updated as follows:

17

$$\begin{array}{c}\hfill C(t+1)=\text{Mod}\left(a,e^{c(t)},\left(1,-,e^{c(t)}\right),,,1\right),\end{array}$$where C(0) is set as 1.5 and a is a constant subject to [1.475,4]. Because the exponential-logic-modulo map has a better performance and randomness, the population diversity has been improved. As a result, high-quality solutions are easier to obtain and improve the convergence of the algorithm.

Cauchy mutation (NHBBA2)

The Cauchy mutation is derived from the Cauchy distribution [38], where the one-dimensional Cauchy probability density function concentrated near the origin, which is defined as Eq. (18). Cauchy mutation is an efficient technique used to improve optimization algorithms [38, 39]. Figure 2 shows the curves of the standard Cauchy density function and the standard Gauss density function. Based on the distribution of the Cauchy “heavy tail” property, i.e. a long tail, the Cauchy mutation has a wider distribution than the Gaussian mutation produces random numbers with a wider range of distribution, so the Cauchy mutation is stronger than the Gaussian mutation in terms of perturbation. If the Cauchy mutation is used in the individual velocity update of the algorithm in place of the Gaussian mutation to generate offspring, the algorithm is more likely to jump out of the local optimum solution. The Cauchy mutation in the binary algorithm is shown by Eq. (19):

18

$$\begin{array}{cc}\hfill f_t(v)=& \frac{1}{\pi }\frac{t}{t^2+v^2},\hfill \end{array}$$

19

$$\begin{array}{cc}\hfill v_k^{t+1}=& v_k^t+v_k^t\ast \text{cauchy}(0,1),\hfill \end{array}$$

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Fig. 2

Probability density distribution curves for the standard Gauss and Cauchy distributions

Chaotic map (NHBBA3)

In the field of optimisation, chaotic maps can be used to generate chaotic numbers between 0 and 1 instead of pseudo-random number generators due to its characteristics of non-repetition, nonlinear and long-term unpredictability [24, 40]. It has been experimentally demonstrated that using chaotic sequences for population initialization, selection, crossover, and mutation affects the entire algorithm’s process and frequently achieves better results than pseudo-random numbers [41, 42].

In BBA, pulse emission rate r and loudness A are the important parameters that affect the performance of the BBA. First, r is a gradually increasing sequence, while A is a gradually decreasing sequence, as seen in Eqs. (13) and (12). However, only when the condition $\mathrm{rand}<A(i)$ is met, a new solution could be obtained in the BBA (see line 13 in the algorithm 1). Because A is a parameter that decreases monotonically, the likelihood that a new solution would be accepted decreases with each iteration. As a result, some workable solutions can be overlooked. Chaotic map is introduced to replace r and A to boost algorithm diversity in order to address this issue.

The pseudo-code of NHBBA algorithm is shown in Algorithm 2. In Algorithm 2, $\mathrm{\Delta }$ is constant, set to 0.7 in this paper.

[See PDF for image]

Fig. 3

Flowchart for UC problem using NHBBA

NHBBA for solving UC problem

The flowchart of UC problem using the proposed binary bat algorithm is shown in Fig. 3. The major procedure of Fig. 3 and Algorithm 2 have displayed the algorithm’s implementation steps for solving UC problem. For heuristic constraint handling in Fig. 3, the spinning reserve algorithm, up/down time constraint procedure and unit de-commitment algorithm are taken from [43]. The following steps are the explanation of Fig. 3.

• Step1: Some initial parameters about loudness $A_k$, pulse emission rate $r_k$, minimum frequency $F_{\min }$ and maximum frequency $F_{\max }$ and the maximum number of iteration are set.

• Step2: Initial ON/OFF solution with size $N\times T$ are generated randomly [48]. Then, verify that the initial state satisfies the minimum up/down time constraints, load constraints at each time slots. If they do proceed to the next step, otherwise commit or de-commit the required time slots and units.

• Step3: Calculate the fitness values included fuel cost and start-up cost using Eq. (3).

• Step4: For each individual, the frequency and velocity are updated by Eq. (8) and (9).

• Step5: If each $r_k<\mathrm{\Delta }$, the crossover operator is applied to update velocity by Eq. (16), otherwise velocity is update by Eq. (19).

• Step6: Map the bat position to binary space by Eq. (14) and (15).

• Step7: The constraint handling algorithm [43] is applied to repair constraints (4), (5), (6) and (7) because the position update procedure in step 5 may cause the individual to be unfeasible.

• Step8: Calculate the fitness values included fuel cost and start-up cost using Eq. (3).

• Step9: The values of loudness $A_k$ and pulse emission rate $r_k$ are changed by chaotic map.

• Step10: Repeat step 4 to step 9 until the maximum number of evaluations.

Table 1. Description of partly IEEE CEC 2017 benchmark functions

No.	Functions	Class	Range
F1	Shifted and rotated Bent Cigar function	Unimodal	[−100,100]
F2	Shifted and rotated Zakharov function	Unimodal	[−100,100]
F3	Shifted and rotated Rosenbrock’s function	Multimodal	[−100,100]
F4	Shifted and rotated Rastrigin’s function	Multimodal	[−10,10]
F6	Shifted and rotated Lunacek Bi-Rastrigin function	Multimodal	[−20,20]
F12	Hybrid function 3 (N=3)	Hybrid	[−100,100]
F13	Hybrid function 4 (N=4)	Hybrid	[−100,100]
F14	Hybrid function 5 (N=4)	Hybrid	[−100,100]
F21	Composition function 1 (N=3)	Composition	[−100,100]
F22	Composition function 2 (N=3)	Composition	[−100,100]

Experimental results and discussion

A significant number of experiments were conducted to evaluate the performance of the proposed NHBBA. Firstly, ten benchmark functions in CEC 2017 [44] are employed to compare NHBBA with four other binary optimization algorithm. The benchmark functions including unimodal, multimodal, hybrid, and composition functions are presented in Table 1. All comparison algorithms were tested in the same hardware environment to guarantee fairness in the experiment. The parameter sensitive analysis and computational complexity of NHBBA are also examined and explained. Finally, a power system’s UC problems are tested and analysed to validate the effectiveness of NHBBA in engineering application problems. The bold in Tables 2–4 indicate the optimal results for different frequencies, population sizes, and Max iteration in the representation function, respectively. The bold in Table 5 indicates the corresponding optimal values of “Ave”, “Std” and “Time” for each benchmark function in the table.

Parameter sensitive analysis

This section mainly focus on the parameter sensitive analysis of the proposed NHBBA. The required parameters of the NHBBA are loudness A, frequency F, pulse emission rate r, population size N and maximum iterations $N_{\mathrm{gen}}$ from Algorithm 2. A and r are replaced by chaotic maps in this paper. The specific analysis for the parameters are shown in Section 4.3. The choices for the other parameters are as follows.

Table 2. Frequency analysis of representative functions for NHBBA

F	F1		F4		F12		F22
	Ave	Std	Ave	Std	Ave	Std	Ave	Std
[0, 0.5]	1.47E+03	5.18E+02	6.32E+01	1.02E+01	1.17E+03	2.61E+02	3.67E+03	8.20E+02
[0, 1]	1.41E+03	4.25E+02	6.10E+01	1.31E+01	1.05E+03	3.78E+02	3.55E+03	1.04E+03
[0, 1.5]	1.30E+03	5.75E+02	6.03E+01	1.23E+01	1.03E+03	3.39E+02	3.25E+03	1.10E+03
[0, 2]	1.22E+03	1.69E+02	5.68E+01	1.04E+01	8.69E+02	3.24E+02	2.76E+03	1.26E+03
[0, 2.5]	1.38E+03	5.94E+02	5.95E+01	1.15E+01	1.02E+03	2.52E+02	3.40E+03	1.45E+03

From the definition of Equation 9 and 14, it can been found that frequency F has an influence on the convergence speed of the proposed algorithm. To analyse the effect of different frequency on the proposed NHBBA, we set a group of frequency intervals, such as [0,0.5], [0,1], [0,1.5], [0,2], and [0,2.5], according to the range of frequency in the original BBA [34]. The unimodal function F1, multimodal function F4, hybrid function F12, and composition function F22 are chosen as representative functions to analyse the effect of each frequency interval. In addition, each representative function gives an average fitness value Ave and standard deviation Std resulted from 50 population size, 500 iterations, and 30 independent runs. Table 2 shows the frequency analysis of representative functions. From Table 2, the frequency interval [0,2] obtains the optimal in Ave. From the longitudinal direction, Ave tends to increase regardless of whether the frequency interval on either side of [0,2] is widened or narrowed. Therefore, [0,2] can be recommended as the best frequency interval for the proposed algorithm.

Table 3. Population size analysis of representative functions

N	F1		F4		F12		F22
	Ave	Std	Ave	Std	Ave	Std	Ave	Std
10	1.63E+03	5.81E+02	6.92E+01	1.27E+01	1.19E+03	3.65E+02	3.84E+03	1.33E+03
20	1.43E+03	4.74E+02	6.14E+01	9.46E+00	1.11E+03	3.37E+02	3.58E+03	1.22E+03
30	1.22E+03	1.69E+02	5.68E+01	1.04E+01	8.69E+02	3.24E+02	2.79E+03	1.26E+03
40	1.29E+03	5.91E+02	5.88E+01	7.69E+00	9.71E+02	2.33E+02	3.31E+03	1.08E+03
50	1.29E+03	4.29E+02	5.86E+01	1.03E+00	9.27E+02	2.36E+02	2.76E+03	1.07E+03

To enhance algorithm’s search efficiency, it is important to determine the population size. Yang’s study demonstrated that a population size between 15 and 50 can be applied to most problems [34]. While it is beneficial to have a large population to increase population and avoid local optimal, it can take more time to find the best individuals in the current iteration. Thus, to determine the optimal population size of the proposed NHBBA, the population size levels were determined as 10, 20, 30, 40, and 50, and then 30 independent experiments are conducted the unimodal function F1, multimodal function F4, hybrid function F12, and composition function F22 at 500 iterations. Table 3 reveals that the proposed NHBBA with 30 population sizes achieved the most favourable results for F1, F4, F12 functions, but there is a slightly less favourable outcome for function F22. Besides, it can be seen that the search performance of the proposed NHBBA is gradually enhanced as the population size grows from 10 to 30. This means that it limits the algorithm’s search ability when the population size is too small. However, when the population size is between 30 and 50, the algorithms achieve very close results, which means that it is difficult for individuals to obtain valid information in the population, even if the population is enlarged. Obviously, it is undeniable that continuously increasing the population size increases the computational efficiency of the algorithm. Therefore, population size was set as 30 in this paper.

Table 4. Max iteration analysis of representative functions for NHBBA

Iter	F1		F4		F12		F22
	Ave	Std	Ave	Std	Ave	Std	Ave	Std
100	1.64E+03	5.37E+02	6.44E+01	1.16E+01	1.11E+03	3.81E+02	3.61E+03	1.41E+03
200	1.40E+03	5.16E+02	5.68E+01	1.22E+01	9.00E+02	2.73E+02	2.98E+03	1.06E+03
500	1.22E+03	1.69E+02	5.68E+01	1.04E+01	8.69E+02	3.24E+02	2.76E+03	1.26E+03
1000	1.24E+03	4.62E+02	5.84E+01	1.07E+01	9.45E+02	3.00E+02	2.79E+03	1.23E+03

The experimental results are also influenced by the number of algorithm evaluations. Typically, when the number of iterations is larger, the algorithm optimization yields better results. A suitable maximum number of iterations is essential due to the time cost and stochastic nature of the algorithm In order to select a maximum number of iterations suitable for NHBBA, four sets of maximum iterations such as 100, 200, 500, and 1000 were tested independently for 30 times using the unimodal function F1, multimodal function F4, hybrid function F12, and composition function F22. The population size of this experiment is 50. From Table 4, it can be seen that F1, F4, F12, and F22 obtain the best average fitness value in 30 independent runs with a maximum iteration of 500. However, the standard deviation of F12 and F22 is slightly lower in 500 maximum iterations. Besides, the average fitness of the 1000 maximum iterations is slightly worse than that of the 500 maximum iterations, which may be due to the chaotic map exacerbating the randomness of the algorithm. In terms of the results, the maximum number of 500 iterations is suited for the NHBBA.

Algorithm computational complexity

Computational complexity of the proposed NHBBA is discussed in this section. The computational complexity of the NHBBA can be divided into seven steps in this paper: initialization, fitness evaluation, frequency update, velocity update, transfer function, the improved crossover operator, and chaotic map. The time consumption for the initialization is $O(N\ast D)$. The fitness function depends heavily on the optimization process, it needs to take $O(T\ast N\ast D)$ times. The time consumption for frequency update and velocity update is $O(T\ast N)$ and $O(T\ast N\ast D)$. It needs $O(v\ast T)$ times for transfer function. The time consumption for the crossover operator is $O(T\ast N\ast D)$. The time consumption for chaotic map is $O(c\ast T)$. N is the number of bats, D is the function’s dimension, T is the maximum iteration of times the algorithm need to evaluate. In addition, v and c are simply denoted as the times required for a single transfer function and chaotic map, respectively. Overall, the computational complexity of the proposed NHBBA is $O(N\ast D+3\ast T\ast N\ast D+3\ast T(N+v+c))$.

Performance of NHBBA for CEC 2017 benchmark functions

[See PDF for image]

Fig. 4

Convergence curves of the NHBBA and other algorithms on representative functions

This experiment aims to validate the effective of the proposed algorithm compared to BBA [34],binary dragonfly algorithm (BDA) [45], binary grey wolf optimizer (BGWO) [46], and binary Harris hawks optimizer (BHHO) [47] in benchmark functions. All of the tests in this section are conducted under the same conditions, in order to maintain fairness. The population size is set to 30. The maximum iteration is set uniformly to 500, and all algorithms test the benchmark function 30 times to reduce the weight of random settings. The compared algorithms’ additional parameters are derived from the original paper. The comparison experiment results of NHBBA with the other algorithm on CEC 2017 are presented in Table 5. In Table 5, $\mathrm{Ave}$, $\mathrm{Std}$ and $\mathrm{Time}$ are average fitness value, standard deviation and convergence time under 30 independent runs.

Table 5. Comparison results with the other algorithms

Algorithm	Item	F1	F2	F3	F4	F6
NHBBA	Ave	1.22E+03	1.33E+03	1.28E+03	5.68E+01	1.32E+02
	Std	1.69E+02	1.59E+02	3.60E+02	1.04E+01	1.27E+02
	Time	3.01E+00	3.63E+00	3.10E+00	3.62E+00	3.71E+00
BBA	Ave	2.26E+03	2.16E+03	2.17E+03	8.09E+01	1.46E+02
	Std	6.89E+02	8.86E+02	8.43E+02	1.77E+01	1.82E+01
	Time	1.48E+00	1.62E+00	1.47E+00	1.29E+00	1.36E+00
BDA	Ave	1.14E+04	1.02E+04	1.02E+04	1.97E+02	6.05E+02
	Std	3.10E+03	2.73E+03	3.26E+03	4.15E+01	1.60E+02
	Time	3.02E+00	3.25E+00	9.53E+00	2.77E+00	2.78E+00
BGWO	Ave	1.09E+04	9.98E+03	9.23E+03	1.09E+02	2.06E+02
	Std	2.92E+03	5.62E+03	2.44E+03	3.03E+01	1.19E+02
	Time	2.69E+00	2.79E+00	2.63E+00	2.44E+00	2.53E+00
BHHO	Ave	3.74E+03	3.42E+03	3.18E+03	9.83E+01	2.17E+02
	Std	1.02E+03	1.18E+03	6.45E+02	1.37E+01	4.12E+01
	Time	2.95E+00	3.11E+00	2.86E+00	2.54E+00	2.67E+00
Algorithm	Item	F12	F13	F14	F21	F22
NHBBA	Ave	8.69E+02	1.92E+07	2.01E+01	2.79E+03	9.33E+07
	Std	3.24E+02	1.43E+07	1.79E-01	1.26E+03	7.08E+07
	Time	3.67E+00	3.71E+00	3.57E+00	3.50E+00	3.59E+00
BBA	Ave	1.32E+03	2.80E+07	2.04E+01	4.72E+03	3.06E+08
	Std	3.69E+02	2.33E+07	8.60E-02	1.80E+03	2.08E+08
	Time	1.32E+00	1.36E+00	1.32E+00	1.29E+00	1.38E+00
BDA	Ave	1.31E+04	2.18E+09	2.09E+01	3.58E+04	1.75E+10
	Std	3.23E+03	1.20E+09	1.62E-01	9.56E+03	1.18E+10
	Time	2.79E+00	2.82E+00	9.02E+00	2.76E+00	2.85E+00
BGWO	Ave	4.40E+03	4.47E+08	2.01E+01	1.54E+04	5.04E+09
	Std	1.94E+03	3.62E+08	9.41E-02	7.45E+03	4.01E+09
	Time	2.43E+00	2.49E+00	2.48E+00	2.52E+00	2.67E+00
BHHO	Ave	2.85E+03	1.11E+08	2.04E+01	9.39E+03	1.12E+09
	Std	7.61E+02	5.23E+07	9.41E-02	2.37E+03	4.88E+08
	Time	2.60E+00	2.67E+00	2.53E+00	2.52E+00	2.70E+00

From Table 5, the optimization values obtained by NHBBA are best among all functions. Specifically, the optimization results obtained by NHBBA are all much smaller compared to the results obtained by BBA, especially in F12 and F22. This demonstrates the strong exploration capability of NHBBA. Also, the results obtained by NHBBA are all better than that of BDA, BGWO and BHHO. Besides, the Std obtained by NHBBA performs admirably. Although the Std obtained by NHBBA performs slightly worse in F14 and F6, this does not imply that NHBBA is poorly robust, but rather implies that there may be a certain amount of randomness in the NHBBA that avoids local convergence. Although the convergence time of NHBBA is slightly longer, the optimization results obtained by NHBBA are the best. This is mainly due to the fact that the improved crossover operator not only enhances the diversity of the population and improves the exploration capacity of the algorithm, but also increases the computational complexity.

To exhibit the specific changes in NHBBA and compared algorithms in a more visually and figurative manner, the convergence curves of NHBBA and compared algorithms are shown in Fig. 4. As seen in the figures, the number of convergence steps of F1, F2, F3, F4, F6, F12, F13, F14, F21, and F22 is 125, 90, 272, 173, 163, 149, 236, 135, 226, and 183, respectively. While the corresponding number of convergence steps obtained by BBA are about 94, 2, 89, 151, 175, 272, 330, 103, 79, and 5, respectively. As a result, the convergence steps of NHBBA on the hybrid function, including F12, F13, and F14, are significantly smaller than that of BBA. Besides, no matter which benchmark function is tested, the optimal solution obtained by NHBBA is obviously superior to BBA and the other three compared algorithms. Although the convergence process of NHBBA is not the best when comparing the test function with the other four algorithms, it has the best optimal solutions.

Experiments on unit commitment

The effectiveness of the proposed NHBBA is verified to solve UC problem with 10-unit, 20-unit, 40-unit and IEEE 118-bus test system for 24-h scheduling horizon. The required spinning reserve for these test systems are considered as 10% of the load demand. The comparison of both proposed approaches with other approaches demonstrates the effectiveness of NHBBA in terms of GA [8], LR [8], SA [9], LRGA [50], MA [51], DPLR [52], GACC [53], BFWO [54], BGWO [55], hGADE/cur1 [56], HHSRSA [57], BFMO [21], ABFMO [21], BCS [58], BDEr [59], BPSOGWO [60], MFO [61], and BAMFO [62] , etc., for 10-unit, 20-unit, and 40-unit test system. To evaluate the performance of the proposed algorithm, BBA is tested on each test system as a comparison of NHBBA. For a fair comparison, population sizes, max iterations, minimum frequency $F_{\min }$ and maximum frequency $F_{\max }$ of both BBA and NHBBA are 30, 500, 0 and 2, respectively. In addition to the parameters, the loudness A and pulse rate r of BBA are set as 0.6 and 0.7, respectively. Due to the randomness of these algorithms, both BBA and NHBBA run independently 30 times. To verify the validity of each innovation point, four test units are tested in NHBBA1, NHBBA2 and NHBBA3.

Table 6. Test system data for 10-unit

Unit	$P_{\max }$	$P_{\min }$	$\alpha (\mathrm{\$}/{\mathrm{MW}}^2)$	$\beta (\mathrm{\$}/\mathrm{MWh})$	$\gamma (\mathrm{\$}/h)$	${\mathrm{MUT}}_k$	${\mathrm{MDT}}_k$	${\mathrm{HSC}}_k$	${\mathrm{CSC}}_k$	${\mathrm{CSH}}_k$	$T_0^k$
1	455	150	1000	16.19	0.00048	8	8	4500	9000	5	8
2	455	150	970	17.26	0.00031	8	8	5000	10000	5	8
3	130	20	700	16.6	0.002	5	5	550	1100	4	− 5
4	130	20	680	16.5	0.00211	5	5	560	1120	4	− 5
5	162	25	450	19.7	0.00398	6	6	900	1800	4	− 6
6	80	20	370	22.26	0.00712	3	3	170	340	2	− 3
7	85	25	480	27.74	0.00079	3	3	260	520	2	− 3
8	55	10	660	25.92	0.00413	1	1	30	60	0	− 1
9	55	10	665	27.27	0.00222	1	1	30	60	0	− 1
10	55	10	670	27.79	0.00173	1	1	30	60	0	− 1

Table 7. Load demand for 24-h for 10-unit

Hour	1	2	3	4	5	6	7	8	9	10	11	12
Demand	700	750	850	950	1000	1100	1150	1200	1300	1400	1450	1500
Hour	13	14	15	16	17	18	19	20	21	22	23	24
Demand	1400	1300	1200	1050	1000	1100	1200	1400	1300	1100	900	800

[See PDF for image]

Fig. 5

Load demand pattern

[See PDF for image]

Fig. 6

Convergence characteristics of different test system for NHBBA and BBA

Table 8. Best solutions for 10-unit test problem in 30 runs (Power: Mw, Cost: $/h, Time: s)

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
U1	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455
U2	245	295	370	455	390	360	410	455	455	455	455	455	455	455	455	310	260	360	455	455	455	455	315	345
U3	0	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	0	0	0
U4	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	0	0	0
U5	0	0	25	40	25	25	25	30	85	162	162	162	162	85	30	25	25	25	30	162	85	25	0	0
U6	0	0	0	0	0	0	0	0	20	33	73	80	33	20	0	0	0	0	0	33	20	20	0	0
U7	0	0	0	0	0	0	0	0	25	25	25	25	25	25	0	0	0	0	0	25	25	25	0	0
U8	0	0	0	0	0	0	0	0	0	10	10	43	10	0	0	0	0	0	0	10	0	0	0	0
U9	0	0	0	0	0	0	0	0	0	0	10	10	0	0	0	0	0	0	0	0	0	0	0	0
U10	0	0	0	0	0	0	0	0	0	0	0	10	0	0	0	0	0	0	0	0	0	0	0	0

Cost = 563937.6875    Time = 18.6037

Table 9. Best solutions for 20-unit test problem in 30 runs (Power: Mw, Cost: $/h, Time: s)

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
U1	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455
U2	245	295	382.5	455	455	425	455	455	455	455	455	455	455	455	455	310	260	360	437.5	455	455	415	432.5	345
U3	0	0	0	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	0	0	0	0
U4	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	0	0
U5	0	0	25	40	25	25	45	30	97.5	162	162	162	162	97.5	30	25	25	25	25	162	150	25	25	0
U6	0	0	0	0	0	0	0	0	20	33	73	80	33	20	0	0	0	0	0	33	20	20	0	0
U7	0	0	0	0	0	0	0	0	25	25	25	25	25	25	0	0	0	0	0	25	25	25	0	0
U8	0	0	0	0	0	0	0	0	0	10	10	43	10	0	0	0	0	0	0	10	0	0	0	0
U9	0	0	0	0	0	0	0	0	0	0	10	10	0	0	0	0	0	0	0	0	0	0	0	0
U10	0	0	0	0	0	0	0	0	0	0	0	10	0	0	0	0	0	0	0	0	0	0	0	0
U11	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455
U12	245	295	382.5	455	455	425	455	455	455	455	455	455	455	455	455	310	260	360	437.5	455	455	415	432.5	345
U13	0	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	0	0
U14	0	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	0	0
U15	0	0	0	40	25	25	45	30	97.5	162	162	162	162	97.5	30	25	25	25	25	162	150	0	0	0
U16	0	0	0	0	0	0	0	0	20	33	73	80	33	20	0	0	0	0	20	33	20	0	0	0
U17	0	0	0	0	0	0	0	0	0	25	25	25	25	0	0	0	0	0	25	25	25	0	0	0
U18	0	0	0	0	0	0	0	0	0	10	10	43	10	0	0	0	0	0	0	0	0	0	0	0
U19	0	0	0	0	0	0	0	0	0	0	10	10	0	0	0	0	0	0	0	10	0	0	0	0
U20	0	0	0	0	0	0	0	0	0	0	0	10	0	0	0	0	0	0	0	0	0	0	0	0

Cost = 1123559.639    Time = 43.6543

Table 10. Best solutions for 40-unit test problem in 30 runs (Power: Mw, Cost: $/h, Time: s)

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
U1	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455
U2	245	295	388.75	443.75	455	455	455	455	455	455	455	455	455	455	455	342.5	292.5	386.25	455	455	455	397.5	395	278.33
U3	0	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130
U4	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	0	0
U5	0	0	25	25	25	27.5	45	30	103.75	162	162	162	162	103.75	50	25	25	25	45	162	123.75	25	25	0
U6	0	0	0	0	0	0	0	0	20	33	73	80	33	20	20	0	0	0	0	33	20	20	0	0
U7	0	0	0	0	0	0	0	0	25	25	25	25	25	0	0	0	0	0	25	25	25	0	0	0
U8	0	0	0	0	0	0	0	0	0	10	10	43	10	0	0	0	0	0	0	10	0	0	0	0
U9	0	0	0	0	0	0	0	0	0	0	10	10	0	0	0	0	0	0	0	0	0	0	0	0
U10	0	0	0	0	0	0	0	0	0	0	0	10	0	0	0	0	0	0	0	0	0	0	0	0
U11	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455
U12	245	295	388.75	443.75	455	455	455	455	455	455	455	455	455	455	455	342.5	292.5	386.25	455	455	455	397.5	395	278.33
U13	0	0	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130
U14	0	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	0	0	0	0
U15	0	0	0	25	25	27.5	45	30	103.75	162	162	162	162	103.75	50	25	25	25	45	162	123.78	25	25	0
U16	0	0	0	0	0	0	0	0	20	33	73	80	33	20	20	0	0	0	0	33	20	20	0	0
U17	0	0	0	0	0	0	0	0	0	25	25	25	25	25	0	0	0	0	25	25	25	0	0	0
U18	0	0	0	0	0	0	0	0	0	10	10	43	10	0	0	0	0	0	0	10	0	0	0	0
U19	0	0	0	0	0	0	0	0	0	0	10	10	0	0	0	0	0	0	0	0	0	0	0	0
U20	0	0	0	0	0	0	0	0	0	0	0	10	0	0	0	0	0	0	0	0	0	0	0	0
U21	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455
U22	245	295	388.75	443.75	455	455	455	455	455	455	455	455	455	455	455	342.5	292.5	386.25	455	455	455	397.5	395	278.33
U23	0	0	0	0	0	0	0	130	130	130	130	130	130	130	0	0	0	0	0	130	130	130	130	130
U24	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130
U25	0	0	0	25	25	27.5	45	30	103.75	162	162	162	162	103.75	50	25	25	25	45	162	123.75	25	0	0
U26	0	0	0	0	0	0	0	0	20	33	73	80	33	20	0	0	0	0	0	33	20	20	0	0
U27	0	0	0	0	0	0	0	0	0	25	25	25	25	0	0	0	0	0	0	25	25	25	0	0
U28	0	0	0	0	0	0	0	0	0	10	10	43	10	0	0	0	0	0	0	10	0	0	0	0
U29	0	0	0	0	0	0	0	0	0	0	10	10	0	0	0	0	0	0	0	0	0	0	0	0
U30	0	0	0	0	0	0	0	0	0	0	0	10	10	0	0	0	0	0	0	0	0	0	0	0
U31	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455	455
U32	245	295	388.75	443.75	455	455	455	455	455	455	455	455	455	455	455	342.5	292.5	386.25	455	455	455	397.5	0	0
U33	0	0	0	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	0	0	0
U34	0	0	0	0	0	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	0	0
U35	0	0	0	0	25	27.5	45	30	103.75	162	162	162	162	103.75	50	25	25	25	45	162	123.75	25	25	25
U36	0	0	0	0	0	0	0	0	20	33	73	80	33	20	0	0	0	0	20	33	0	20	0	0
U37	0	0	0	0	0	0	0	0	0	25	25	25	25	0	0	0	0	25	25	25	0	0	0	0
U38	0	0	0	0	0	0	0	0	0	10	10	43	0	0	10	0	0	0	0	10	0	0	0	0
U39	0	0	0	0	0	0	0	0	0	0	10	10	0	0	0	0	0	0	0	0	0	0	0	0
U40	0	0	0	0	0	0	0	0	0	0	0	10	0	0	0	0	0	0	0	0	0	0	0	0

Cost = 2246675.698    Time = 102.0875

Performance of NHBBA for 10-unit test problem

Table 11. Comparison results for 10-unit test system (Cost: $/h, Time: s)

Approach	Best	Average	Worst	Ave. time
LR [8]	565825	565825	565825	–
DPLR [52]	564049	564049	564049	108
GA [8]	565825	–	570032	221
SA [9]	565828	565988	566260	3
MFO [61]	564810	566093	568520	20.45
LRGA [50]	564800	564800	564800	518
GACC [53]	563977	564791	565606	85
BFWA [54]	563977	564018	564855	65.42
BGWA [55]	563976	564378	565518	–
hGADE/cur1 [56]	563959	564088	564350	24
HHSRSA [57]	563937	563965	563995	16.83
BFMO [21]	–	564864	–	–
ABFMO [21]	–	565136	–	–
BAMFO [62]	563938	563974	563977	17.78
BPSOGWO [21]	–	565210	–	–
BDEr [21]	–	564549	–	–
SABA [63]	563937.7	564311.15	565073.24	29
BBA	563937.68	563963.75	563977.01	13.36
NHBBA1	563937.68	563967.84	563977.01	20.25
NHBBA2	563937.68	563966.52	563977.01	20.96
NHBBA3	563937.68	563957.35	563977.01	18.79

${}^1$ NHBBA1 considers only the improved crossover operator

${}^2$ NHBBA2 considers improved crossover operators and Cauchy mutation

${}^3$ NHBBA3 considers all innovation points

10-unit test system is employed to verify the superiority of the proposed NHBBA in small-scale power system. The load demand and characteristic of 10-unit test problem are taken from [49] and are shown in Tables 7 and 6.

Table 8 represents the best results for 10-unit test problem in 30 runs. The best commitment scheduling for 24 h, the best cost solution ($563937.6875) and corresponding algorithm execution time (18.6037 s) can be obtained from Table 8. Table 11 shows the comparison statistical data for 10-unit test system in 30 runs. Best, Average, Worst, and Ave.time are best value, average value, worst value, and average computation time for 10-unit test problem and in 30 independently runs. The letter “–” indicates that the values are not available in the literature. From Table 11, it can be seen that NHBBA3 performs best cost in 10-unit test system, which $\mathrm{Best}$, $\mathrm{Average}$, and Worst are $563937.68, $563957.35, and $563977.01. Specifically, the cost of NHBBA1 and NHBBA2 is marginally inferior when compared to the simulation results of BBA in 10 units, but NHBBA3 performs the best. This shows that the improvement of NHBBA3 is manifested in the small-scale power system UC problem. The insignificant improvement of NHBBA3 compared to the cost value of BBA may be due to the fact that their cost value are already very close to the optimal solution in the 10-unit test system of this methodology. NHBBA3 consumes less $111.32 and $1887.32 per day compared to LR and DPLR, which indicates that NHBBA outperforms the compared numerical optimization methods. Comparing with other heuristic algorithms, NHBBA3 is the best in terms of optimal, average and worst costs. In addition, the average computation time of NHBBA3 outperforms the numerical methods and other heuristic algorithms excepting SA, HHSRSA, BAMFO, and BBA. This means that the computational complexity of NHBBA is low compared to the computational complexity of other hybrid heuristic algorithms.

Figure 6 displays the total cost convergence characteristics of NHBBA and BBA for 10-unit test problem. It can be achieved that the convergence performance of NHBBA in 10-unit power system is better than that of BBA, which is mainly because the improved NHBBA enhances the population diversity, expands the search space and increases the convergence of NHBBA.

Performance of NHBBA for 20-unit test problem

Table 12. Comparison results for 20-unit test system (Cost: $/h, Time: s)

Approach	Best	Average	Worst	Ave. time
LR [8]	1130660	1130660	1130660	–
DPLR [52]	1128098	–	–	299
GA [8]	1126243	–	1132059	733
SA [9]	1126251	1127955	1129112	17
MA [51]	1128192	1128213	1128403	113
LRGA [50]	1122622	–	–	1147
GACC [53]	1125516	1127153	1128790	225
BFWA [54]	1124658	1124941	1125087	–
BGWA [55]	125546.4	1126126.3	1127393.2	80.47
hGADE/cur1 [56]	1123426	1124502	1125076	48
HHSRSA [57]	1124889	1124912.8	1124951.5	–
BFMO [21]	–	1131958	–	–
ABFMO [21]	–	1131551	–	–
BAMFO [62]	1123825	1124759	1125495	26.22
BPSOGWO [21]	–	1145016	–	–
BDEr [21]	–	1132763	–	–
BBA	1138192.16	1140247.14	1142286.86	26.20
NHBBA1	1124304.67	1124601.35	1126112.42	36.53
NHBBA2	1124059.45	1124514.07	1125543.25	42.28
NHBBA3	1123559.63	1124373.41	1125743.85	45.64

${}^1$ NHBBA1 considers only the improved crossover operator

${}^2$ NHBBA2 considers improved crossover operators and Cauchy mutation

${}^3$ NHBBA3 considers all innovation points

For 20-unit test problem, the 10-unit test system generation data are copied, and the load demand data are multiplied by 2, which is shown in Fig. 5. This test power system’s schedule periods are 24 h.

Table 9 shows the optimal commitment and generation scheduling achieved by NHBBA. The optimal cost is $1123559.639, and corresponding execution time is 43.6543 s. The effectiveness of the proposed NHBBA is evaluated in comparison with the other algorithms in Table 12. The letter “–” indicates that the values are not available in the literature.

It can be obtained from Table 12 that the best cost, average cost, and worst cost of NHBBA3 are $1123559.63, $1124373.41, and $1125743.85, respectively. Compared to NHBBA1 and NHBBA2, the improvements of best cost and average cost of NHBBA3 are significant. This is mainly due to the fact that the improved crossover operator increases the diversity of the population and improves the exploration capability of the algorithm. Meanwhile, Cauchy mutation and chaotic map are effectively avoided the risk of local optimum. In addition, 13.9% of average cost was saved by NHBBA3 compared to BBA. This demonstrates the clear superiority of the NHBBA over the BBA in terms of performance. Compared to LR and DPLR of the numerical methods, NHBBA3 saved $7100.37 and $4538.37 on the best and average costs per day. The average computation time of DPLR is nearly five times as large as that of NHBBA3. Compared with other heuristic algorithms, NHBBA3 significantly performs well in terms of computational time and cost. Although NHBBA3 is not the best one in the experiment results of all the comparison algorithms, it is still in the top ranks.

Figure 6 shows the cost convergence characteristics of NHBBA and BBA for 20-unit test problem. It can be seen that NHBBA has far better convergence and accuracy than BBA, because the crossover operator and chaotic map improve the algorithm population multiplicity, expand the search space and enhance the convergence of the algorithm; the Cauchy mutation effectively avoids the risk of premature convergence of NHBBA in the face of complex problems.

Performance of NHBBA for 40-unit test problem

Table 13. Comparison results for 40-unit test system (Cost: $/h, Time: s)

Approach	Best	Average	Worst	Ave. time
LR [8]	2258503	2258503	2258503	–
DPLR [52]	2256195	2256195	2256195	1200
GA [8]	2251911	–	2259706	2697
SA [9]	2250063	2252125	2254539	88
MA [51]	2249589	2249589	2249589	217
GACC [53]	2249715	2253270	2256824	614
BFWA [54]	2248228	2248572	2248645	238.02
BGWA [55]	2252475	2257866	2263333	169.24
HHSRSA [57]	2248508	2248653	2248757	179.666
BFMO [21]	–	2267669	–	–
ABFMO [21]	–	2265867	–	–
BAMFO [62]	2247165	2249309	2251438	43.27
BPSOGWO [21]	–	2390940	–	–
BDEr [21]	–	2291992	–	–
BBA	2300141.21	2305271.15	2309299.13	41.10
NHBBA1	2247567.61	2249095.67	2249799.90	86.29
NHBBA2	2246937.21	2249078.73	2250152.55	98.05
NHBBA3	2246675.69	2248912.80	2250142.40	105.04

${}^1$ NHBBA1 considers only the improved crossover operator

${}^2$ NHBBA2 considers improved crossover operators and Cauchy mutation

${}^3$ NHBBA3 considers all innovation points

For 40-unit test problem, the 10-unit test system generation data are copied, and the load demand data is multiplied by 4, which is shown in Fig. 5. This test power system’s schedule periods are also 24 h.

Table 10 shows the optimal commitment and generation scheduling achieved by NHBBA. The optimal cost is $2246675.698, and corresponding execution time is 102.0875 s. Table 13 shows the performance comparison of the NHBBA in comparison with other algorithms. The letter “–” indicates that the values are not available in the literature.

From Table 13, it is observed that the best cost is $2246675.69 obtained by NHBBA3. Although the average cost of NHBBA3 is not the smallest, it is only $340.8 more than the BFWA and outperforms the other comparison algorithms. In comparison with the optimal costs of BBA, NHBBA1 and NHBBA2, NHBBA3 saves $53,465.52, $891.92, and $261.52 per day, which indicates that NHBBA is significantly improved in exploration and exploitation compared to the original BBA, and it also means that each improvement of NHBBA enhances the algorithm. Compared to LR and DPLR, $9590.2 and $7282.2 were saved by NHBBA3 per day, respectively. And average computation time of DPLR is nearly 11 times as large as NHBBA3. In comparison of Sects 5.4.1, 5.4.2, and 5.4.3, it can be seen that NHBBA has a smaller cost and shorter computation time compared to numerical methods, as the dimensions of the unit grow. The NHBBA also performs well compared to the other heuristic algorithm. In the 40-unit average cost, NHBBA saves $43079.2, $142027.2, $396.2, and $16954.2 over the novel BDEr, BPSOGWO, BAMFO, and BFMO, respectively. After the above analysis, NHBBA can effectively solve the UC problem in 40-unit test system.

Figure 6 shows the cost convergence characteristics of NHBBA and BBA for 40-unit test problem. In Fig. 6, NHBBA is superior to BBA in both convergence and optimal solution.

Performance of NHBBA for IEEE 118-bus test problem

Table 14. Comparison of results for IEEE 118-bus test system (Cost: $/h, Time: s)

Approach	Best	Average	Worst	Std	Ave. time
BPSO [43]	1655236.925	1662826.954	1671494.185	4752.3526	74.109
BBA	1653515.485	1657588.1	1662875.157	2599.743	114.056
NHBBA1	1626077.602	1626150.298	1626223.693	37.392	131.580
NHBBA2	1626076.872	1626144.473	1626219.177	35.641	148.944
NHBBA3	1626073.128	1626139.166	1626209.885	32.971	152.448

In order to demonstrate the robustness of NHBBA, IEEE 118-bus test system with 54 generating units is considered. The generator data can be found online at [64]. The load is shown in Fig. 5.

Figure 6 describes the convergence characteristic of NHBBA and BBA for IEEE 118-bus test system. From the figure, a better global best solution is obtained by NHBBA and NHBBA converges significantly faster than BBA. In Fig. 6, the performance of NHBBA is better than that of BBA with the increasing complexity of power system.

Table 14 shows the comparison results of NHBBA with BBA and BPSO in terms of IEEE 118-bus test system. According to Table 14, it can be concluded that the total cost of NHBBA is significantly lower than that of BBA and BPSO, with an average cost savings of $31,433.889 and $36,672.743 when compared to BBA and BPSO, respectively. Further, the standard deviation of NHBBA is much smaller than that of BBA and BPSO, indicating that NHBBA is much better at solving UC problems for complex power system than BBA and BPSO. In general, although the average computation time has increased, both the quality of the solution and the robustness of the algorithm have improved.

Concluding remarks

In this paper, a hybrid binary bat algorithm NHBBA has been proposed, which integrates improved operator crossover, Cauchy mutation, and chaotic map into binary bat algorithm. The effectiveness and convergence of the proposed algorithm are analysed and verified by four typical benchmark functions. Meanwhile, we successfully applied NHBBA to solve UC problem and validated the superiority of using NHBBA to deal with four case test systems, compared to binary bat algorithm, numerical methods and other heuristic algorithm. It is important to note that the issue of unit commitment involving renewable energy systems needs to be addressed urgently as more renewable energy sources are integrated into the power system, and this issue will be left for future work. Furthermore, the proposed algorithm is also suitable to deal with different applications such as feature selection and mixed integer linear programming.

Author contributions

AP and HL conceived the methodology; CL, HL, and LY verified the validity of the method; AP wrote the main manuscript text, and CL and HL reviewed and edited the manuscript. All authors reviewed the manuscript.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 62163013 and in part by the National Natural Science Foundation of Hubei Province under Grant 2021CFB542.

Declarations

Conflict of interest

The authors declared that they have no conflicts of interest to this work.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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