Optimal Scheduling Study of Hydro–Solar Complementary System Based on Improved Beluga Whale Algorithm

Abstract

The optimization scheduling model of the hydro–solar complementary system has the characteristics of high dimension, nonlinearity, strong constraints, etc., and it is difficult to solve. In view of this problem, this paper proposes an Improved Beluga Whale Optimization to solve the model. The local development strategy of the IBWO is replaced by the spiral movement of the whale algorithm to enhance the local development ability of the algorithm. In addition, an elimination mechanism is added after the whale fall stage of the original algorithm to increase the population diversity and improve the ability of the algorithm to jump out of the local optimum. This paper compares the solution effect of the IBWO algorithm with several well-known algorithms on 24 classic test functions and 29 CEC2017 test functions; the superior performance of the IBWO algorithm is verified. With the maximum power generation as the goal, the power generation scheduling model of the Beipan River hydro–solar complementary system is constructed and solved by the BWO algorithm, the IBWO algorithm, and the SCA algorithm, respectively. The results show that the IBWO algorithm can effectively improve the power generation of the hydro–solar complementary system and has a faster convergence speed than the BWO algorithm and the SCA algorithm, providing a new optimization tool for dealing with complex engineering optimization problems.

1. Introduction

In the context of “carbon peak, carbon neutral” and “building a new type of power system with new energy as the main body” and other major national strategies, the use of abundant water and solar resources to build a new type of green and low-carbon power system and improve the level of clean energy utilization and operation efficiency of the power system is the way to achieve the goal of “double carbon”. Integrating photovoltaic and hydropower, utilizing the fast adjustment ability of hydraulic turbines to hedge against the fluctuation of wind and solar output, and forming a high-quality and stable complementary power generation system can effectively promote new energy consumption on the grid, and improve the power generation and peak shifting ability in the operation of hydro–solar multi-energy complementary systems [1]. However, with the increase in the types of power sources, the objectives and constraints of the scheduling body also change, and the solution of the optimal scheduling problem of the hydro–solar complementary system becomes more and more complex. The objective of a hydro–solar complementary system is to find the optimal scheduling solution that maximizes the total amount of power generated by the whole system while satisfying a set of physical constraints, which is a complex multivariate, nonlinear, and strongly constrained problem. Over the past decades, many attempts have been made to find effective methods, and the existing methods can be broadly categorized into mathematical planning methods and meta-heuristic algorithms based on the search principles employed. The former methods are represented by linear programming [2], nonlinear programming [3] and dynamic programming [4]. The latter methods are mainly grouped into four categories: evolution-based algorithms, population-based algorithms, physics-based algorithms, and human-based algorithms. Evolution-based algorithms include genetic algorithms [5], differential evolution [6], co-evolutionary algorithms [7], etc.; population-based algorithms include particle swarm algorithms [8], Harris hawk algorithms [9], gray wolf algorithm [10], cuckoo algorithm [11], etc.; physics-based algorithms include the simulated annealing algorithm [12], gravitational search algorithm [13], lightning search algorithm [14], etc.; human-based algorithms include the harmony search algorithm [15], teaching learning-based optimization algorithm [16], etc. Some new meta-heuristic algorithms have emerged in recent years, such as the snake algorithm [17], the horse flock algorithm [18], and the Kepler optimization algorithm [19].

In the field of cascade reservoir optimization and scheduling, meta-heuristic algorithms have attracted the attention of many scholars due to their powerful optimization capabilities. Nguyen et al. [20] proposed an adaptive selective cuckoo search algorithm and used it to solve three short-term hydro-thermal power dispatching problems. Yin et al. [21] proposed a crisscross optimization (CSO) algorithm to solve the short-term hydro-thermal power dispatching problem. Feng et al. [22] proposed a new multi-strategy gravitational search algorithm (MGSA) to solve the ecological dispatch problem of the Wu River hydropower system. Niu et al. [23] proposed a hybrid quantum particle swarm optimization algorithm and used it to solve the joint dispatch problem of cascade reservoirs.

According to the no free lunch theorem, there is no one algorithm suitable for all optimization problems and each algorithm has its advantages and disadvantages [24]. The Beluga Whale Optimization (BWO) algorithm was proposed by Zhong et al. [25]. The development of the BWO algorithm was inspired by the behavior of beluga whales including swimming, feeding, and landing, and the BWO algorithm has been successfully applied to industrial design. Yuan et al. [26] proposed a hybrid BWO algorithm based on a jellyfish search optimizer (HBWO-JS) and demonstrated the usefulness of the HBWO-JS algorithm in three real-world engineering designs and two truss topology optimization problems. Hussien et al. [27] adopted three improvement strategies based on the original BWO algorithm and solved eight engineering problems.

Targeting the disadvantages of the algorithm, such as low convergence accuracy and it being easy to fall into the local optimum, this paper proposes an improved beluga optimization algorithm. The main contributions of this study are as follows: (1) The algorithm replaces the local development strategy in the original algorithm with a spiral motion to improve the search ability of the algorithm, and uses an elimination mechanism to improve the ability of the algorithm to jump out of the local optimum. (2) In order to verify the reliability of the improved algorithm, the algorithm is tested on 24 classic test functions and 29 CEC2017 test functions. The results show that the IBWO algorithm is highly competitive with other comparison algorithms and the improvement strategy is effective. (3) The IBWO algorithm is used to solve the optimal scheduling problem of the hydro–solar complementary system of the Beipan River in China. The results show that the IBWO algorithm has good engineering practicability.

The rest of this paper is organized as follows. Section 2 briefly describes the details of the BWO algorithm and the proposed algorithm. The performance of the IBWO algorithm in the test function is verified in Section 3. Section 4 describes the objective, the constraints, and the way to handle the constraints of the hydro–solar complementary model. Section 5 takes the Beipan River hydro–solar complementary system as an example to verify the engineering practicability of the IBWO algorithm. Finally, conclusions are given in Section 6.

2. The Proposed Method

2.1. Beluga Whale Optimization (BWO)

The BWO algorithm consists of three phases: exploration phase, exploitation phase, and whale fall. Beluga whales are considered as individuals in the population. The BWO algorithm can be modeled according to the equilibrium factor $B_f$ transition from exploration to exploitation:

$$B_f=B_0(1-M/2M_{\max })$$

(1)

where $M$ is the current iteration, $M_{\max }$ is the maximum iterative number, and $B_0$ randomly changes between (0, 1) at each iteration. The exploration phase happens when the balance factor $B_f>0.5$ while the exploitation phase happens when $B_f\le 0.5$.

2.1.1. Exploration Phase

The exploration phase of the BWO algorithm is built by considering the swim behavior of beluga whale, whose position is updated as follows.

$$\left\{\begin{array}{l}{X}_{i,j}^{M+1}=X_{i,pj}^M+(X_{r,p1}^M-X_{i,pj}^M)\times (1+r_1)\times \sin (2\pi r_2),\hspace{1em}j=even\\ X_{i,j}^{M+1}=X_{i,pj}^M+(X_{r,p1}^M-X_{i,pj}^M)\times (1+r_1)\times \cos (2\pi r_2),\hspace{1em}j=odd\end{array}\right.$$

(2)

where $X_{i,j}^{M+1}$ is the new position for the ith beluga whale on the jth dimension, $p_j(j=1,2,\cdots ,d)$ is a random number selected from d-dimension, $X_{i,pj}^M$ is the position of the ith beluga whale on $p_j$ dimension, $X_{i,pj}^M$ and $X_{r,p1}^M$ are the current positions for the ith and rth beluga whale (r is a randomly selected beluga whale), and $r_1$ and $r_2$ are random numbers between (0, 1).

2.1.2. Exploitation Phase

The strategy of Levy flight [28] is introduced in the exploitative phase of BWO to enhance the convergence, and the mathematical model is expressed as follows:

$$X_i^{M+1}=r_3{X}_{best}^M-r_4{X}_i^M+C_1\times L_F\times (X_r^M-X_i^M)$$

(3)

$$C_1=2r_4(1-M/M_{\max })$$

(4)

where $X_i^M$ and $X_r^M$ are the current position for the ith beluga whale and a random beluga whale, $X_i^{M+1}$ is the new position of the ith beluga whale, $X_{best}^M$ is the best position among the beluga whales, and $r_3$ and $r_4$ are random numbers between (0, 1).

$L_F$ is the Levy flight function [28], calculated as follows:

$$L_F=0.05\times {\displaystyle \frac{u\times \sigma }{{\left|v\right|}^{1/\beta }}}$$

(5)

$$\sigma ={\left({\displaystyle \frac{\Gamma (1+\beta )\times \sin (\pi \beta /2)}{\Gamma ((1+\beta )/2)\times \beta \times 2^{(\beta -1)/2}}}\right)}^{1/\beta }$$

(6)

where $u$ and $v$ are normally distributed random numbers and $\beta$ is the default constant equal to 1.5.

2.1.3. Whale Fall

The positions of beluga whales and step size of whale fall are used to establish the updated position. The mathematical model is expressed as follows:

$$X_i^{M+1}=r_5{X}_i^M-r_6{X}_r^M+r_7{X}_{step}$$

(7)

where $r_5$, $r_6$ and $r_7$ are random numbers between (0,1) and $X_{step}$ is the step size of whale fall established as follows:

$$X_{step}=(u_b-l_b)\exp (-C_{2}M/M_{\max })$$

(8)

$$C_2=2W_f\times Num$$

(9)

$$W_f=0.1-0.05M/M_{\max }$$

(10)

where $Num$ is the population size, $W_f$ is the probability of whale fall, and $u_b$ and $l_b$ are the upper and lower boundary of variables, respectively.

2.2. Proposed Method

2.2.1. Spiral Motion

The beluga whale algorithm has low convergence accuracy, and the exploitation strategy of the beluga algorithm is replaced by a spiral motion with a stronger local search capability, which is formulated as follows:

$$\overrightarrow{X}(M+1)={\overrightarrow{D}}^{\prime }\times e^{bl}\times \cos (2\pi l)+{\overrightarrow{X}}^{\ast }(M)$$

(11)

$$a_2=-1+M\times ((-1)/M_{\max })$$

(12)

$$l=(a_2-1)\times rand+1$$

(13)

where ${\overrightarrow{D}}^{\prime }=\left|{\overrightarrow{X}}^{*}(M)-\overrightarrow{X}(M)\right|$ is the distance from the ith individual to the optimal solution obtained so far and $b$ is a constant that defines the shape of the logarithmic spiral; the value of $b$ in this paper is 1.

2.2.2. Elimination Mechanism

The standard beluga algorithm makes it easy to fall into the local optimum. Adding an elimination strategy after the whale-fall phase of each iteration involves performing Gaussian mutation on the optimal solution at the current iteration, the formulas are shown in (14) and (15), to obtain N mutant solutions, and all the current solutions being sorted in ascending order of fitness, eliminating the N solutions with the worst fitness and replace them with mutant solutions, makes it easier for the algorithm to jump out of the local optimum; the value of N in this paper is 5.

$$f(x)={\displaystyle \frac{1}{\sqrt{2\pi \sigma }}}\exp (-{\displaystyle \frac{{(x-\mu )}^2}{2{\sigma }^2}})$$

(14)

$$X_{i,j}=X_{best,j}\times (1+f(x))$$

(15)

$X_{i,j}$ is the jth dimension of the ith individual, $X_{best,j}$ is the optimal solution at the current iteration, and $f(x)$ is a standard normally distributed random number.

2.3. Framework of IBWO

The flowchart of the proposed IBWO algorithm is shown in Figure 1, and the execution steps are as follows:

Step 1: Initialize the algorithm parameters and randomly generate the initial population in the search space.

Step 2: Calculate the fitness values of all individuals in the initial population and obtain the optimal individual and the corresponding fitness value.

Step 3: Determine whether the algorithm enters the exploration phase or the exploitation phase based on the balance factor $B_f$. If $B_f$ > 0.5, the algorithm enters the exploration phase and uses Formula (2) to update the individual position. If $B_f$ ≤ 0.5, the algorithm enters the exploitation phase and uses Formula (11) to update the individual position.

Step 4: Enter the whale fall phase and use Formula (7) to update the individual position.

Step 5: Use the elimination mechanism to update the individual position according to Formula (14).

Step 6: Check whether the algorithm’s iterative termination condition is met. If so, output the optimal solution. Otherwise, return to step 3 to continue the loop.

2.4. Computational Complexity

The computational complexity of the proposed IBWO algorithm is the same as that of the original BWO algorithm except for the elimination mechanism. The computational complexity of the elimination mechanism and BWO algorithm is $O(5\times M_{\max })$ and $O(n\times (1+1.1\times M_{\max }))$, respectively. The computational complexity of IBWO algorithm is evaluated as $O(n\times (1+1.1\times M_{\max })+5\times M_{\max })$.

3. Numerical Experiments to Verify the Performance of the IBWO Method

3.1. Benchmark Functions Set I

3.1.1. Classic Test Functions

In order to evaluate the performance of the IBWO algorithm, the algorithm performance is tested using 24 classic test functions (F1–F24), which have been widely used in the literature [29]. These problems are categorized into unimodal and multimodal, the unimodal test functions (F1–F9) reveal the exploitation performance and the multimodal test functions (F10–F24) challenge the exploration capability, the details of classic test functions are referenced in the literature [25].

For the classic test functions, the algorithm termination criterion is the maximum number of iterations, which is set as 500, and the population size is set as 50, and for each function, each algorithm is run independently 30 times to obtain its statistical results. In addition, in the next experiments, the best results shown in the tables are highlighted in bold in order to facilitate the analysis of the results.

3.1.2. Parameter Settings

In order to demonstrate the superiority of the proposed algorithm, several well-known evolutionary algorithms are selected for comparison, including the sine cosine algorithm (SCA) [30], moth flame algorithm (MFO) [31], arithmetic average approach (AOA) [32], whale algorithm (WOA) [33], prairie dog algorithm (PDO) [34], and reptile algorithm (RSA) [35]. The parameters used for all the algorithms are shown in

Table 1. Parameter settings for each algorithm.

Algorithm	Parameter	Value
SCA	a	2
MFO	r	[−1, −2]
	b	1
AOA	α	5
	μ	0.5
WOA	a	[0, 2]
	b	1
PDO	ρ	0.1
	ε	2.2204 × 10−16
	Δ	0.005
RSA	α	0.1
	β	0.005
BWO	$W_f$	[0.1, 0.05]
IBWO	$W_f$	[0.1, 0.05]
	P	0.9

.

All the experiments are implemented on environment of Microsoft Windows 11 (64-bit) on Intel (R) Core (TM) i9-13900HX CPU with 2.20 GHz and 32.0 GB (RAM), and the programming software is MATLAB R2022b.

3.1.3. Experimental Results

Table 2. Experimental results of mean and standardized values of all algorithms on classic test functions.

Function	Metrics	SCA	MFO	AOA	WOA	PDO	RSA	BWO	IBWO
F1	mean	3.2242E+00	1.3352E+03	7.6977E-17	4.0616E-84	0.0000E+00	0.0000E+00	1.5616E-260	0.0000E+00
	std	1.0548E+01	4.3426E+03	4.2162E-16	1.3089E-83	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00
	rank	5	6	4	3	1	1	2	1
F2	mean	6.1054E-03	3.0420E+01	0.0000E+00	1.0724E-53	0.0000E+00	0.0000E+00	1.2409E-132	0.0000E+00
	std	7.7817E-03	1.6471E+01	0.0000E+00	4.7709E-53	0.0000E+00	0.0000E+00	4.0593E-132	0.0000E+00
	rank	4	5	1	3	1	1	2	1
F3	mean	2.7574E-04	7.4043E-11	0.0000E+00	2.7967E-125	0.0000E+00	0.0000E+00	1.7939E-268	0.0000E+00
	std	1.4679E-03	2.2222E-10	0.0000E+00	1.4117E-124	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00
	rank	5	4	1	3	1	1	2	1
F4	mean	6.5723E+03	1.5910E+04	8.0211E-04	2.5703E+04	0.0000E+00	0.0000E+00	1.1366E-243	0.0000E+00
	std	4.2141E+03	1.1468E+04	2.3477E-03	9.3763E+03	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00
	rank	4	5	3	6	1	1	2	1
F5	mean	2.9048E+01	5.9273E+01	2.0937E-02	4.1845E+01	0.0000E+00	0.0000E+00	3.3556E-127	0.0000E+00
	std	1.0695E+01	8.6989E+00	2.0075E-02	2.9555E+01	0.0000E+00	0.0000E+00	1.7852E-126	0.0000E+00
	rank	4	6	3	5	1	1	2	1
F6	mean	9.7224E+03	1.2939E+04	2.8113E+01	2.7359E+01	1.5831E+01	1.4434E+01	5.0851E-07	0.0000E+00
	std	3.2669E+04	3.0859E+04	3.6536E-01	3.4013E-01	1.3555E+01	1.4682E+01	8.6442E-07	0.0000E+00
	rank	7	8	6	5	4	3	2	1
F7	mean	1.0192E+01	2.0349E+03	2.8802E+00	7.3462E-02	2.8800E+00	6.6892E+00	5.4244E-15	0.0000E+00
	std	9.3800E+00	5.5346E+03	2.5169E-01	7.4832E-02	1.7797E+00	8.3102E-01	1.0420E-14	0.0000E+00
	rank	7	8	5	3	4	6	2	1
F8	mean	7.1233E-02	2.4413E+00	4.8053E-05	2.4605E-03	6.5014E-05	8.2912E-05	5.4864E-05	1.5437E-04
	std	6.0893E-02	4.9077E+00	4.1995E-05	3.1039E-03	5.6623E-05	9.5413E-05	4.8615E-05	1.2198E-04
	rank	7	8	1	6	3	4	2	5
F9	mean	2.1605E+01	2.9727E+02	2.5190E+02	4.9563E+02	0.0000E+00	0.0000E+00	2.5765E-229	0.0000E+00
	std	1.2379E+01	1.0254E+02	6.9551E+01	1.0795E+02	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00
	rank	3	5	4	6	1	1	2	1
F10	mean	−3.8421E+03	−8.7436E+03	−5.5681E+03	−1.1475E+04	−3.8941E+03	−5.4649E+03	−1.2569E+04	−1.2569E+04
	std	2.9956E+02	8.5520E+02	3.7483E+02	1.3621E+03	3.0733E+02	2.3343E+02	1.5377E-09	1.8501E-12
	rank	7	3	4	2	6	5	1	1
F11	mean	5.4649E+00	4.3542E+00	0.0000E+00	5.1502E-01	6.9728E-01	0.0000E+00	0.0000E+00	0.0000E+00
	std	1.6102E+00	9.1082E-01	0.0000E+00	6.2531E-01	1.8217E+00	0.0000E+00	0.0000E+00	0.0000E+00
	rank	5	4	1	2	3	1	1	1
F12	mean	−5.9748E+02	−1.0176E+03	−5.0816E+02	−1.1170E+03	−9.3361E+02	−5.1794E+02	−1.1750E+03	−1.1750E+03
	std	3.6289E+01	3.5747E+01	5.9751E+01	8.3461E+01	4.2913E+01	3.9949E+01	4.8559E-12	2.2737E-13
	rank	5	3	7	2	4	6	1	1
F13	mean	3.3682E+01	1.5923E+02	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00
	std	2.9614E+01	3.4366E+01	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00
	rank	2	3	1	1	1	1	1	1
F14	mean	1.1265E+01	1.0235E+01	4.4409E-16	5.0626E-15	4.4409E-16	4.4409E-16	4.4409E-16	4.4409E-16
	std	9.4945E+00	9.2496E+00	0.0000E+00	2.4948E-15	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00
	rank	4	3	1	2	1	1	1	1
F15	mean	8.0157E-01	2.1818E+01	1.0701E-01	2.1114E-02	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00
	std	2.5551E-01	5.1185E+01	7.3064E-02	4.6353E-02	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00
	rank	4	5	3	2	1	1	1	1
F16	mean	2.7228E-10	1.9924E-12	4.7285E-08	−1.6667E-01	−1.0000E+00	−1.0000E+00	−1.0000E+00	−1.0000E+00
	std	1.2556E-10	3.6208E-12	3.4877E-08	3.7905E-01	0.0000E+00	0.0000E+00	0.0000E+00	0.0000E+00
	rank	4	3	5	2	1	1	1	1
F17	mean	2.0158E+02	4.3588E+00	4.3839E-01	8.6896E-03	5.3793E-01	1.2737E+00	1.8681E-14	1.5705E-32
	std	7.9299E+02	2.8094E+00	5.0238E-02	8.7275E-03	5.1242E-01	3.0668E-01	2.8914E-14	5.5674E-48
	rank	8	7	4	3	5	6	2	1
F18	mean	1.3525E+04	9.5031E+00	2.7996E+00	2.1295E-01	2.8969E+00	2.6829E-01	9.4598E-14	1.3498E-32
	std	3.6220E+04	6.5449E+00	1.1654E-01	1.3747E-01	2.8863E-01	8.2246E-01	1.5943E-13	5.5674E-48
	rank	8	7	5	3	6	4	2	1
F19	mean	1.3303E+00	1.7882E+00	8.6437E+00	1.6212E+00	5.2960E+00	3.0180E+00	9.9800E-01	2.1653E+00
	std	7.5136E-01	1.6468E+00	4.3019E+00	1.8611E+00	4.6418E+00	1.7497E+00	2.6520E-09	3.5616E+00
	rank	2	4	8	3	7	6	1	5
F20	mean	1.0837E-03	9.7486E-04	1.5746E-02	7.2820E-04	2.7656E-03	1.3767E-03	3.2782E-04	3.1790E-04
	std	3.2028E-04	3.2140E-04	2.3789E-02	4.7497E-04	1.0234E-02	6.9838E-04	1.6331E-05	1.5488E-05
	rank	5	4	8	3	7	6	2	1
F21	mean	−1.0316E+00	−1.0316E+00	−1.0316E+00	−1.0316E+00	−1.0316E+00	−1.0307E+00	−1.0315E+00	−1.0316E+00
	std	2.7246E-05	6.7752E-16	1.0834E-07	2.2169E-10	4.0975E-08	1.3510E-03	1.0661E-04	5.1992E-16
	rank	1	1	1	1	1	3	2	1
F22	mean	−2.5803E+00	−7.3070E+00	−3.9336E+00	−8.3693E+00	−5.4930E+00	−5.0552E+00	−1.0148E+01	−1.0153E+01
	std	1.8810E+00	3.2042E+00	1.2507E+00	2.5934E+00	2.7533E+00	3.2046E-07	6.3122E-03	5.5785E-15
	rank	8	4	7	3	5	6	2	1
F23	mean	−3.9253E+00	−8.3170E+00	−4.1836E+00	−8.8437E+00	−6.5014E+00	−5.0877E+00	−1.0396E+01	−1.0403E+01
	std	1.8462E+00	3.2676E+00	1.4766E+00	2.6510E+00	2.9598E+00	9.4738E-07	1.1673E-02	4.7728E-13
	rank	8	4	7	3	5	6	2	1
F24	mean	−4.4769E+00	−7.8330E+00	−4.2609E+00	−8.4461E+00	−5.4637E+00	−5.1285E+00	−1.0529E+01	−1.0536E+01
	std	1.4970E+00	3.6280E+00	1.7322E+00	3.0951E+00	3.1768E+00	1.4018E-06	1.1849E-02	1.7337E-11
	rank	7	4	8	3	5	6	2	1
mean rank		5.17	4.75	4.08	3.13	3.13	3.25	1.67	1.33
final ranking		7	6	5	3	3	4	2	1

shows the experimental results which include mean (Ave) and standard deviation (Std) of all algorithms on the classic test functions. First, we analyze the results of unimodal functions (F1–F9) which are problems with no local optimum but only global optimum. From Table 2, it is obvious that the IBWO algorithm is the most competitive algorithm, achieving the best results on 8 of 9 problems (F1–F7, F9), and the results show that the spiral motion improves the individual’s search ability.

Next, we focus on the results of multimodal functions (F10–F24). Multimodal functions are problems with a large number of local optimal solutions, and for such functions, the key question we are most interested in is whether the algorithm can jump out of the local optimum. As shown in Table 2, the IBWO algorithm is also the best algorithm, achieving the best results on 14 of 15 evaluation problems (F10–F18, F20–F24), and the results show that the elimination strategy improves the algorithm’s ability to jump out of local optima.

On the classic test functions, we can find that IBWO obtains 22 first rankings and 2 fifth rankings, and its average ranking is 1.33, which is the first overall ranking. Figure 2 shows the convergence curves of eight algorithms on 24 classic test functions. The results show that the convergence speed of the IBWO algorithm is highly competitive.

Table 3. The running time of each algorithm (second).

Function	Metrics	SCA	MFO	AOA	WOA	PDO	RSA	BWO	IBWO
F1	mean	0.0572	0.0589	0.0478	0.0425	0.5892	0.4929	0.0986	0.0924
	rank	3	4	2	1	8	7	6	5
F2	mean	0.0571	0.0605	0.0490	0.0443	0.5791	0.4788	0.1000	0.0926
	rank	3	4	2	1	8	7	6	5
F3	mean	0.1059	0.1146	0.0980	0.1011	0.6031	0.5113	0.1468	0.1106
	rank	3	5	1	2	8	7	6	4
F4	mean	0.1443	0.1455	0.1336	0.1428	0.6744	0.5745	0.1988	0.2872
	rank	3	4	1	2	8	7	5	6
F5	mean	0.0579	0.0587	0.0475	0.0453	0.5740	0.4818	0.0983	0.0890
	rank	3	4	2	1	8	7	6	5
F6	mean	0.0673	0.0713	0.0591	0.0612	0.5874	0.4966	0.1114	0.1155
	rank	3	4	1	2	8	7	5	6
F7	mean	0.0581	0.0606	0.0470	0.0494	0.5795	0.4878	0.0994	0.0917
	rank	3	4	1	2	8	7	6	5
F8	mean	0.1191	0.1203	0.1109	0.1362	0.6073	0.5161	0.1669	0.2308
	rank	2	3	1	4	8	7	5	6
F9	mean	0.0714	0.0707	0.0602	0.0572	0.5894	0.5000	0.1128	0.1159
	rank	4	3	2	1	8	7	5	6
F10	mean	0.0725	0.0689	0.0605	0.0596	0.5845	0.4945	0.1089	0.1148
	rank	4	3	2	1	8	7	5	6
F11	mean	0.0730	0.0694	0.0536	0.0575	0.5871	0.4870	0.1051	0.1045
	rank	4	3	1	2	8	7	6	5
F12	mean	0.1291	0.1279	0.1182	0.1272	0.6452	0.5502	0.1798	0.2550
	rank	4	3	1	2	8	7	5	6
F13	mean	0.0665	0.0665	0.0479	0.0455	0.5768	0.4830	0.0998	0.0931
	rank	4	3	2	1	8	7	6	5
F14	mean	0.0745	0.0720	0.0511	0.0495	0.5839	0.4966	0.1011	0.0981
	rank	4	3	2	1	8	7	6	5
F15	mean	0.0780	0.0796	0.0614	0.0602	0.5871	0.4949	0.1105	0.1182
	rank	3	4	2	1	8	7	5	6
F16	mean	0.0888	0.0836	0.0704	0.0772	0.5931	0.4991	0.1177	0.1287
	rank	4	3	1	2	8	7	5	6
F17	mean	0.2283	0.2320	0.2174	0.2315	0.7426	0.6547	0.2833	0.4700
	rank	2	4	1	3	8	7	5	6
F18	mean	0.2359	0.2379	0.2118	0.2365	0.7513	0.6586	0.2871	0.4704
	rank	2	4	1	3	8	7	5	6
F19	mean	0.2928	0.2959	0.2928	0.3339	0.3555	0.3217	0.3460	0.6561
	rank	2	3	1	5	7	4	6	8
F20	mean	0.0232	0.0244	0.0227	0.0403	0.1085	0.0825	0.0490	0.0635
	rank	2	3	1	4	8	7	5	6
F21	mean	0.0223	0.0241	0.0222	0.0388	0.0733	0.0533	0.0467	0.0649
	rank	2	3	1	4	8	6	5	7
F22	mean	0.0304	0.0318	0.0300	0.0511	0.1201	0.0909	0.0580	0.0808
	rank	2	3	1	4	8	7	5	6
F23	mean	0.0336	0.0371	0.0337	0.0570	0.1254	0.0978	0.0622	0.0889
	rank	1	3	2	4	8	7	5	6
F24	mean	0.0400	0.0408	0.0379	0.0607	0.1285	0.1019	0.0670	0.0997
	rank	2	3	1	4	8	7	5	6
mean rank		2.88	3.46	1.38	2.38	7.96	6.83	5.38	5.75
final ranking		3	4	1	2	8	7	5	6

shows the running time of each algorithm on different classic test functions. The results show that the IBWO algorithm takes slightly more time than the BWO algorithm and ranks sixth among all algorithms. In summary, the IBWO algorithm is a superior and competitive algorithm compared to other comparative algorithms.

3.2. Benchmark Functions Set II: IEEE CEC2017

3.2.1. CEC2017 Test Functions

In this section, the CEC2017 test function is chosen to test the performance of the IBWO algorithm. Compared with the classic test functions, the CEC2017 test functions are more complex and more challenging to solve. In addition, the optimal solutions of these two test functions are different, and based on these two test functions, the performance of the IBWO algorithm can be effectively measured. For the CEC2017 test functions, the termination criterion is the maximum number of function evaluations, set to be 10,000×D, where D is the dimension of the test function, and the search range is [−100,100]. The detailed information of the 30 CEC2017 benchmark functions are referenced in the literature [36], which includes unimodal functions (F1–F3), multimodal functions (F4–F10), hybrid functions (F11–F20), and composition functions (F21–F30) [36]. Due to the unstable behavior of the F2 function, it was excluded from this set during actual testing [37]. For the 29 test functions, each algorithm is run independently 30 times to obtain its statistical results. To make it easier to analyze the results, the best results shown in the table are highlighted in bold.

3.2.2. Statistical Results Analysis

Table 4. Statistical results of several methods on 30 variable IEEE CEC2017 test functions.

Function	Metrics	SCA	MFO	AOA	WOA	PDO	RSA	BWO	IBWO
F1	mean	1.21E+10	8.26E+09	4.16E+10	2.56E+06	3.61E+10	4.23E+10	4.43E+10	1.06E+05
	std	1.87E+09	5.71E+09	5.20E+09	2.07E+06	9.65E+09	5.49E+09	3.89E+09	5.22E+04
	rank	4	3	6	2	5	7	8	1
F2		NA	NA	NA	NA	NA	NA	NA	NA
F3	mean	3.54E+04	8.76E+04	7.58E+04	1.43E+05	6.45E+04	7.15E+04	7.14E+04	6.71E+03
	std	8.25E+03	5.44E+04	7.67E+03	4.76E+04	8.45E+03	8.83E+03	4.86E+03	2.56E+03
	rank	2	7	6	8	3	5	4	1
F4	mean	1.38E+03	9.04E+02	9.16E+03	5.51E+02	7.60E+03	8.31E+03	9.67E+03	5.00E+02
	std	2.62E+02	3.67E+02	1.99E+03	4.87E+01	3.46E+03	2.27E+03	1.28E+03	2.47E+01
	rank	4	3	7	2	5	6	8	1
F5	mean	7.77E+02	6.99E+02	8.09E+02	7.97E+02	8.25E+02	8.93E+02	8.92E+02	7.03E+02
	std	2.42E+01	4.67E+01	3.92E+01	7.18E+01	5.08E+01	3.11E+01	1.63E+01	5.09E+01
	rank	3	1	5	4	6	8	7	2
F6	mean	6.48E+02	6.34E+02	6.64E+02	6.67E+02	6.78E+02	6.79E+02	6.82E+02	6.10E+02
	std	5.17E+00	1.06E+01	6.33E+00	8.73E+00	9.65E+00	4.45E+00	5.96E+00	8.97E+00
	rank	3	2	4	5	6	7	8	1
F7	mean	1.12E+03	1.06E+03	1.31E+03	1.26E+03	1.39E+03	1.35E+03	1.34E+03	1.10E+03
	std	3.27E+01	1.64E+02	3.77E+01	7.94E+01	1.66E+02	4.58E+01	2.79E+01	8.81E+01
	rank	3	1	5	4	8	7	6	2
F8	mean	1.05E+03	9.93E+02	1.03E+03	1.01E+03	1.06E+03	1.12E+03	1.12E+03	9.43E+02
	std	2.24E+01	5.24E+01	3.01E+01	5.61E+01	6.11E+01	1.38E+01	1.44E+01	2.56E+01
	rank	5	2	4	3	6	8	7	1
F9	mean	5.03E+03	6.46E+03	5.98E+03	8.04E+03	7.16E+03	9.14E+03	9.96E+03	4.92E+03
	std	8.77E+02	2.24E+03	7.90E+02	2.68E+03	2.29E+03	8.37E+02	8.23E+02	9.09E+02
	rank	2	4	3	6	5	7	8	1
F10	mean	8.08E+03	5.08E+03	6.51E+03	6.18E+03	7.78E+03	7.76E+03	8.22E+03	4.69E+03
	std	3.60E+02	8.61E+02	4.95E+02	7.86E+02	5.15E+02	3.45E+02	2.66E+02	6.44E+02
	rank	7	2	4	3	6	5	8	1
F11	mean	2.23E+03	3.60E+03	3.35E+03	1.51E+03	7.58E+03	6.69E+03	5.37E+03	1.21E+03
	std	4.76E+02	2.85E+03	9.53E+02	1.77E+02	2.95E+03	1.22E+03	7.39E+02	4.37E+01
	rank	3	5	4	2	8	7	6	1
F12	mean	1.15E+09	1.87E+08	8.48E+09	5.33E+07	8.65E+09	1.27E+10	7.49E+09	2.59E+06
	std	2.80E+08	5.18E+08	2.64E+09	3.00E+07	3.02E+09	3.47E+09	1.63E+09	1.81E+06
	rank	4	3	6	2	7	8	5	1
F13	mean	4.07E+08	9.94E+06	4.06E+04	1.40E+05	4.96E+09	8.58E+09	4.03E+09	2.63E+04
	std	1.58E+08	2.47E+07	1.38E+04	8.08E+04	1.68E+09	4.18E+09	1.51E+09	4.09E+04
	rank	5	4	2	3	7	8	6	1
F14	mean	1.58E+05	3.00E+05	5.50E+04	9.19E+05	2.37E+06	2.94E+06	1.13E+06	1.92E+05
	std	1.07E+05	1.19E+06	4.84E+04	8.32E+05	1.84E+06	1.84E+06	5.16E+05	1.50E+05
	rank	2	4	1	5	7	8	6	3
F15	mean	1.43E+07	6.39E+04	2.45E+04	7.96E+04	9.99E+07	5.06E+08	1.05E+08	1.02E+04
	std	1.39E+07	6.89E+04	1.16E+04	6.70E+04	1.21E+08	1.45E+08	4.66E+07	1.04E+04
	rank	5	3	2	4	6	8	7	1
F16	mean	3.65E+03	3.12E+03	3.95E+03	3.60E+03	4.59E+03	4.66E+03	4.93E+03	2.65E+03
	std	1.82E+02	3.97E+02	6.44E+02	4.25E+02	4.73E+02	5.34E+02	2.30E+02	3.43E+02
	rank	4	2	5	3	6	7	8	1
F17	mean	2.40E+03	2.58E+03	2.78E+03	2.49E+03	3.07E+03	4.12E+03	3.34E+03	2.31E+03
	std	1.39E+02	2.60E+02	2.53E+02	2.98E+02	3.71E+02	1.84E+03	2.68E+02	2.51E+02
	rank	2	4	5	3	6	8	7	1
F18	mean	3.52E+06	1.48E+06	1.24E+06	1.93E+06	1.24E+07	4.09E+07	1.52E+07	1.20E+06
	std	2.67E+06	3.24E+06	1.43E+06	1.98E+06	1.14E+07	3.28E+07	1.09E+07	1.46E+06
	rank	5	3	2	4	6	8	7	1
F19	mean	2.06E+07	8.91E+06	1.06E+06	2.19E+06	2.67E+08	5.01E+08	1.64E+08	1.58E+04
	std	1.17E+07	3.50E+07	1.22E+05	1.71E+06	2.05E+08	6.88E+08	7.95E+07	1.71E+04
	rank	5	4	2	3	7	8	6	1
F20	mean	2.59E+03	2.67E+03	2.76E+03	2.77E+03	2.83E+03	2.88E+03	2.80E+03	2.50E+03
	std	1.24E+02	2.54E+02	1.99E+02	1.93E+02	1.22E+02	1.08E+02	1.01E+02	1.95E+02
	rank	2	3	4	5	7	8	6	1
F21	mean	2.55E+03	2.49E+03	2.61E+03	2.58E+03	2.67E+03	2.66E+03	2.68E+03	2.48E+03
	std	2.26E+01	4.32E+01	3.91E+01	5.01E+01	7.68E+01	3.49E+01	3.03E+01	8.89E+01
	rank	3	2	5	4	7	6	8	1
F22	mean	8.43E+03	6.26E+03	7.88E+03	6.91E+03	8.41E+03	7.96E+03	7.86E+03	4.91E+03
	std	2.20E+03	1.73E+03	8.76E+02	1.91E+03	1.31E+03	1.03E+03	4.95E+02	2.06E+03
	rank	8	2	5	3	7	6	4	1
F23	mean	2.99E+03	2.82E+03	3.41E+03	3.04E+03	3.29E+03	3.26E+03	3.23E+03	2.88E+03
	std	2.42E+01	2.73E+01	1.74E+02	9.07E+01	7.30E+01	1.18E+02	3.41E+01	6.71E+01
	rank	3	1	8	4	7	6	5	2
F24	mean	3.16E+03	2.97E+03	3.75E+03	3.17E+03	3.38E+03	3.35E+03	3.45E+03	3.16E+03
	std	2.97E+01	2.76E+01	1.63E+02	9.65E+01	7.12E+01	1.02E+02	5.88E+01	8.71E+01
	rank	2	1	7	3	5	4	6	2
F25	mean	3.20E+03	3.31E+03	4.44E+03	2.95E+03	4.39E+03	4.49E+03	4.17E+03	2.90E+03
	std	7.91E+01	4.37E+02	5.54E+02	3.31E+01	6.34E+02	5.81E+02	1.25E+02	1.99E+01
	rank	3	4	7	2	6	8	5	1
F26	mean	6.84E+03	5.79E+03	9.76E+03	7.61E+03	9.06E+03	9.82E+03	9.72E+03	6.11E+03
	std	4.82E+02	4.03E+02	1.02E+03	1.29E+03	1.14E+03	1.10E+03	5.46E+02	7.51E+02
	rank	3	1	7	4	5	8	6	2
F27	mean	3.40E+03	3.25E+03	4.24E+03	3.36E+03	3.64E+03	3.72E+03	3.76E+03	3.27E+03
	std	4.15E+01	2.15E+01	2.90E+02	1.02E+02	8.03E+01	3.59E+02	9.74E+01	2.35E+01
	rank	4	1	8	3	5	6	7	2
F28	mean	3.79E+03	4.13E+03	6.07E+03	3.30E+03	5.46E+03	6.05E+03	5.87E+03	3.23E+03
	std	1.11E+02	7.18E+02	8.61E+02	4.91E+01	7.46E+02	7.57E+02	2.29E+02	2.27E+01
	rank	3	4	8	2	5	7	6	1
F29	mean	4.65E+03	4.01E+03	5.87E+03	4.95E+03	5.68E+03	5.82E+03	6.06E+03	3.89E+03
	std	2.03E+02	2.39E+02	6.74E+02	5.00E+02	4.51E+02	7.73E+02	4.19E+02	2.83E+02
	rank	3	2	7	4	5	6	8	1
F30	mean	7.14E+07	4.15E+05	2.17E+08	8.90E+06	8.82E+08	2.31E+09	5.46E+08	1.21E+04
	std	2.07E+07	5.90E+05	7.02E+08	6.51E+06	4.29E+08	9.85E+08	2.08E+08	4.65E+03
	rank	4	2	5	3	7	8	6	1
mean rank		3.66	2.76	4.97	3.55	6.07	7.00	6.52	1.28
final ranking		4	2	5	3	6	8	7	1

shows the experimental results of all algorithms on the CEC2017 test functions which include mean (Ave) and standard deviation (Std). On the unimodal functions (F1, F3), the IBWO algorithm achieves the best results on 2 of the 2 test functions (F1, F3). On multimodal functions (F4–F10), the IBWO algorithm achieves the best results on 5 of the 7 test functions (F4, F6, F8–F10). In this paper, hybrid and composition functions are used to evaluate the ability of the IBWO algorithm to jump out of the local optimum, which are considered to be the most challenging optimization problems. On the hybrid functions (F11–F20), the IBWO algorithm achieves the best results on 9 of 10 test functions (F11–F13, F15–F20). On the composition functions (F21–F30), the IBWO algorithm achieves the best results on 6 of 10 test functions (F21–F22, F25, F28–F30). On the CEC2017 benchmark functions, we can find that the IBWO algorithm obtains 22 first rankings, 6 s rankings, 1 third ranking, and its average ranking is 1.28, which is the first overall ranking. Therefore, this experiment proves that IBWO algorithm is competitive with other comparative algorithms and the improvement strategy is effective.

4. IBWO Algorithm for Solving the Optimal Scheduling Problem of Hydro–Solar Complementary System

4.1. Mathematical Model

4.1.1. Objective Function

In this paper, the optimal scheduling model of a hydro–solar complementary system is established with the objective of maximizing the total power generation in the scheduling period [38].

The objective function is as follows:

$$\max F={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^I\left(P_{H,i,t}+P_{PV,i,t}\right)}}\cdot \Delta t$$

(16)

where $P_{PV,i,t}$ is the photovoltaic output of the ith outgoing channel in time period t, unit is 10⁴ kW. $P_{H,i,t}$ is the hydropower output of the ith outgoing channel in time period t, unit is 10⁴ kW. $\Delta t$ is the time period length of time period t, unit is h. $T$ represents the total number of time periods.

4.1.2. Physical Constraints

(1) Water balance constraint

$$V(t+1)=V\left(t\right)+\left(I\left(t\right)-Q_D\left(t\right)\right)\Delta t$$

(17)

where $V(t+1)$ is the end storage capacity of the reservoir in time period t. $V\left(t\right)$ is the beginning storage capacity of the reservoir in time period t, unit is 10⁸ m³. $I\left(t\right)$ is the average inflow of the reservoir in time period t. $Q_D\left(t\right)$ is the average outflow of the reservoir in time period t, unit is m³/s.

(2) Water level constraint

$$Z_{u,i,t}^{\min }\le Z_{u,i,t}\le Z_{u,i,t}^{\max }$$

(18)

where $Z_{u,i,t}$ is the upstream water level of the ith power station in the tth time period, and $Z_{u,i,t}^{\max }$ and $Z_{u,i,t}^{\min }$ are the upper and lower limits of the upstream water level of the corresponding power station in the corresponding time period, unit is m.

(3) Power generation flow constraint

$$Q_{g,i,t}^{\min }\le Q_{g,i,t}\le Q_{g,i,t}^{\max }$$

(19)

where $Q_{g,i,t}$ is the power generation flow. $Q_{g,i,t}^{\max }$ and $Q_{g,i,t}^{\min }$ represent the upper and lower limits of the power generation outflow of the corresponding power station in the corresponding time period, and the unit is m³/s.

(4) Outflow constraint

$$Q_{i,t}\ge Q_{i,t}^{\min }$$

(20)

where $Q_{i,t}$ is the total outflow of the ith power station in the tth time period, which consists of the power generation flow $Q_{g,i,t}$ and abandoned water flow $Q_{a,i,t}$. $Q_{i,t}^{\min }$ represent the lower limits of the total outflow of the corresponding power station in the corresponding time period, and the unit is m³/s.

(5) Output constraint

$$P_{H,i,t}^{\min }\le P_{H,i,t}\le P_{H,i,t}^{\max }$$

(21)

where $P_{H,i,t}$ is the generation output of the ith power station in the tth time period. $P_{H,i,t}^{\max }$ and $P_{H,i,t}^{\min }$ are the upper and lower limits of the generation output of the corresponding power station in the corresponding time period, unit is 10⁴ kW.

(6) Delivery channel constraint

$$P_{H,i,t}+P_{PV,i,t}\le TC{C}_{i,t}$$

(22)

where $TC{C}_{i,t}$ is the capacity limit of the ith hydropower delivery channel at moment t, unit is 10⁴ kW.

(7) Beginning and end water level constraints

$$\left\{\begin{array}{l}{Z}_{i,0}=Z_i^{begin}\\ Z_{i,T}=Z_i^{end}\end{array}\right.$$

(23)

where $Z_i^{begin}$ is the beginning water level of the ith hydropower station and $Z_i^{end}$ is the end water level.

4.2. Details of IBWO for the Optimal Scheduling Problem of Hydro–Solar Systems

4.2.1. Individual Structure and Swarm Initialization

In order to solve the equation and inequality constraints efficiently, the upstream water level of each reservoir is used as a decision variable, and each individual in the IBWO algorithm consists of the upstream water levels of all the reservoirs, which are described as follows.

$$X=\left[\begin{array}{cccc}{Z}_{1,1}& Z_{1,2}& \cdots & Z_{1,T}\\ Z_{2,1}& Z_{2,2}& \cdots & Z_{2,T}\\ ⋮& ⋮& Z_{i,t}& ⋮\\ Z_{Num,1}& Z_{Num,2}& \cdots & Z_{Num,T}\end{array}\right]$$

(24)

In the initialization phase, any element of the solution is randomly generated in the feasible range between the maximum and minimum values.

$$Z_{i,t}=Z_{i,t}^{down}+rand\times (Z_{i,t}^{up}-Z_{i,t}^{down})$$

(25)

where rand is a random number between (0, 1).

4.2.2. Constraint Handling Method

In this paper, two types of constraint processing are used; the specific operations are as follows:

• Constrained corridor

For the optimal scheduling problem, the determination of feasible space is decisive for the acquisition of feasible solutions. If the feasible space is too large, it can easily lead to algorithm time-consuming and difficulty obtaining the optimal solution, while if the feasible space is too small, it can easily lead to low search efficiency. In this paper, we use a constrained corridor processing strategy so that the decision variables are optimized in the feasible space as much as possible. In the optimal scheduling problem of the reservoir, various variable constraints are usually converted into state variables such as water level values according to their corresponding mappings, and then constraint coupling is performed to find the intersection, so it is called constrained corridor [39].

2.

Constrained objective method

On the basis of the constraint corridor, this article adopts the constraint objective method to handle the constraints of the scheduling model, calculates the degree of individual violation of constraints, and uses it as the constraint objective value. When comparing the advantages and disadvantages of individuals, the priority is given to comparing the constraint objective value. Individuals with small constraint objectives are better than those with large constraint objectives. When the constraint objectives are the same or neither violates the constraints, the objective function size of the individuals is compared.

4.3. Overall Implementation Framework

The solution process of the optimal scheduling problem of the hydro–solar complementary system using the IBWO algorithm is as follows:

Step 1: Take the water level constraint as the solution space and initialize the initial population, where the individuals in the population are the upstream water levels of each reservoir and the parameters are initialized.

Step 2: Through the relationship between water level and reservoir capacity curve, the output constraint and flow constraint are converted into water level constraint and the intersection with the original water level constraint is taken to obtain the constraint corridor, thereby reducing the possibility of infeasible solutions. The fitness of all individuals in the initial population, i.e., the objective function value, is calculated, and the constraint target is calculated according to the constraint conditions to obtain the optimal individual in the initial population and the corresponding objective function value and constraint target.

Step 3: Determine whether the algorithm enters the exploration phase or the development phase based on the balance factor $B_f$. If $B_f$ > 0.5, the algorithm enters the exploration phase and uses Formula (2) to update the individual position. If $B_f$ ≤ 0.5, the algorithm enters the development phase and uses Formula (11) to update the individual position. When determining whether the newly generated individual replaces the old individual, the constraint targets of the two are compared first. If the constraint target of the newly generated individual is smaller, it is replaced; otherwise, it is discarded. If the constraint targets of the two are the same or neither violates the constraint, the objective function values of the two are compared. If the objective function of the newly generated individual is larger, it is replaced; otherwise, it is discarded. The same operation is performed after updating the individual position in Step 4 and Step 5.

Step 4: Enter the whale fall stage and use Formula (7) to update the individual position.

Step 5: Use the elimination mechanism to update the individual position according to Formula (14).

Step 6: Determine whether the end condition is met; if so, output the optimal solution; otherwise, return to Step 3 and continue the cycle.

5. Case Study

5.1. Engineering Background

The Beipan River Basin has abundant water, rich hydro energy resources, concentrated water level difference in the main stream, and good topographic and geological conditions. Four cascade reservoirs have been put into operation in the basin, from upstream to downstream, including Shannipo (SNP), Guangzhao (GZ), Mamaya (MMY), and Dongqing (DQ), with a total installed capacity of 2664 MW, and photovoltaic power stations Gangping (GP), Yongxin (YX), and Zhenliang (ZL) have a total installed capacity of 750 MW.

This model selects the measured monthly runoff process and photovoltaic data of the Beipan River cascade reservoirs in 2020 (wet year), 2018 (normal year), and 2016 (dry year) for scheduling calculations of the hydro–solar complementary system. The scheduling horizon is set as 1 year split into 12 months, the reservoir water level is used as the decision variable, and the photovoltaic power station uses the measured data in 2022, which is shown in Figure 3.

In the wet year, the initial water level of SNP is 880.71 m and the final water level is 881.20 m, the initial water level of GZ is 738.43 m and the final water level is 734.38 m, the initial water level of MMY is 582.30 m and the final water level is 582.63 m, and the initial water level of DQ is 487.70 m and the final water level is 488.17 m. In the normal year, the initial water level of SNP is 878.32 m and the final water level is 877.59 m, the initial water level of GZ is 733.34 m and the final water level is 726.06 m, the initial water level of MMY is 582.6 m and the final water level is 582.2 m, and the initial water level of DQ is 487.9 m and the final water level is 487.46 m. In the dry year, the initial water level of SNP is 874.76 m and the final water level is 874.53 m, the initial water level of GZ is 733.94 m and the final water level is 734.71 m, the initial water level of MMY is 583.47 m and the final water level is 583.19 m, and the initial water level of DQ is 488.57 m and the final water level is 488.09 m. The rest of the information is shown in

Table 5. Main parameters of hydropower and photovoltaic in the Beipan River Basin.

Power Station	Type	Installed Capacity (MW)	Normal Water Level (m)	Dead Water Level (m)	Maximum Power Generation Flow (m3/s)	Minimum Generation Flow (m3/s)	Minimum Discharge Flow (m3/s)	Maximum Output (104 kw)	Minimum Output (104 kw)
SNP	hydropower reservoir	185.5	885	865	208.42	0	7	18.55	2.042
GZ	hydropower reservoir	1040	745	691	866	0	27.6	104	18.02
MMY	hydropower reservoir	558	585	580	923.5	0	31	55.8	8.73
DQ	hydropower reservoir	880	490	483	917.2	0	50	88	17.2
GP	photovoltaic power plant	300	—	—	—	—	—	30	—
YX	photovoltaic power plant	300	—	—	—	—	—	30	—
ZL	photovoltaic power plant	150	—	—	—	—	—	15	—

.

5.2. Statistical Result Comparison

In order to compare and analyze the application effect of IBWO in the optimal dispatch of power generation in the hydro–solar complementary system, this paper adopts the IBWO algorithm, the BWO algorithm, and the SCA algorithm to solve the model. For all algorithms, the population size is set as 50, and the maximum number of iterations is set as 5000.

Table 6. Statistical results of several methods in different runoff scenarios (10⁸ kWh).

Runoff	Algorithm	Average Value	Best	Worst	Standard Deviation
Wet year	BWO	100.26	101.12	99.23	0.50
	IBWO	101.83	101.94	101.74	0.09
	SCA	99.32	99.88	98.62	0.31
Normal year	BWO	90.54	90.70	90.38	0.09
	IBWO	91.45	91.49	91.44	0.02
	SCA	89.67	90.30	89.16	0.28
Dry year	BWO	82.43	82.43	82.43	0
	IBWO	82.43	82.43	82.43	0
	SCA	81.11	81.54	80.84	0.23

shows the optimal, average, worst, and standard deviation values of 10 independent runs under three scenarios including wet year, normal year, and dry year. In addition, the details of the default parameter settings for the BWO algorithm, the IBWO algorithm, and the SCA algorithm are consistent with Section 3.1.2.

As can be seen from Table 6, the statistical performance of the IBWO algorithm is better than that of the SCA algorithm in the three runoff scenarios. In both wet and normal years, the statistical performance of the IBWO algorithm is better than that of the BWO algorithm. In the dry year, the statistical performance of the IBWO algorithm is the same as that of the BWO algorithm. The average power generation of the IBWO algorithm in the wet year increased by 157 million kwh or 1.57% compared with the BWO algorithm and by 251 million kwh or 2.53% compared with the SCA algorithm; the average power generation of the IBWO algorithm in the normal year increased by 91 million kwh or 1.01% compared with the BWO algorithm and by 178 million kwh or 1.99% compared with the SCA algorithm; and the average power generation of the IBWO algorithm in the dry year is the same as that of BWO algorithm and increased by 132 million kwh or 1.63% compared with the SCA algorithm. The std values of IBWO are lower than BWO and SCA in all three runoff scenarios, indicating that the IBWO algorithm has better stability in the optimal scheduling problem of the hydro–solar complementary system. Figure 4 shows the average fitness convergence curves. It can be seen from the figure that the IBWO algorithm converges quickly and then remains stable and the optimization result is good. In summary, the application of the IBWO algorithm on the study of optimal scheduling of power generation in the hydro–solar system of the Beipan River proves that the two improvement strategies of replacing the local development strategy with a spiral motion and adopting an elimination strategy effectively enhance the algorithm’s optimization ability to local perturbations, avoiding the individual extremes from falling into the local optimum prematurely, and improving the convergence accuracy, which indicates that the IBWO algorithm, on the problem of optimal scheduling of the hydro–solar complementary system, has engineering practicability, which can improve the power generation capacity of the hydro–solar complementary system.

Figure 5 shows the water level process and output process of the four reservoirs corresponding to the optimal results of the two algorithms. The optimized water level process obtained by the IBWO algorithm can satisfy the water level constraints very well, and in the three runoff scenarios, the water level process of the four reservoirs corresponding to the IBWO algorithm is not lower than that of the BWO algorithm and the SCA algorithm, and higher power generation is obtained by operating with a higher water head.

Figure 6 and Figure 7 show the comparison of power generation and total power generation of each reservoir optimal scheduling system scheme for each reservoir of the hydro–solar complementary system corresponding to the IBWO algorithm, the BWO algorithm, and the SCA algorithm under three runoff scenarios, and it can be found that the power generation in the early stage of the scheduling is less and, with the rise of the water level, the power generation in the later stage is more, which indicates that the obtained scheduling scheme is in line with the actual production of the power station.

Figure 8 shows the outflow process of each reservoir optimized by IBWO algorithm under three runoff scenarios. In the wet year, SNP produces abandoned water in July and September; in the normal year, SNP produces abandoned water in June; in the dry year, SNP produces abandoned water in July. Other reservoirs do not produce abandoned water. The changes of the outflow processes follow the variations of the water levels, the power out, and the power generation.

6. Conclusions

Targeting the disadvantages of the beluga whale algorithm with low convergence accuracy and premature convergence, a new beluga whale algorithm (IBWO) is proposed. In the IBWO algorithm, the local search strategy in the original algorithm is replaced by a spiral motion, and an elimination strategy is used to prompt the algorithm to jump out of the local optimum. The superior performance of the IBWO algorithm is fully proved by the verification using 24 classic test functions and 29 CEC2017 test functions. Applying the IBWO algorithm to the study of optimal scheduling of power generation in the Beipan River hydro–solar complementary system, the results show that the IBWO algorithm is superior to the BWO algorithm and the SCA algorithm, thus providing an effective optimization method for solving complex engineering optimization problems. This paper can provide technical measures to improve the global search ability of the BWO algorithm in global optimization problems. The limitation of the IBWO algorithm is that its running time is not superior to other algorithms. In the future, we can consider developing a mechanism to reduce the running time of the algorithm, and we can consider expanding the algorithm into a multi-objective algorithm to solve the multi-objective scheduling problem of hydro–solar complementary systems.