An improved butterfly optimization algorithm and its application in cascade hydropower generation operation

Hydropower is a widely used clean energy. With the growing construction of reservoirs in recent decades, cascade hydropower reservoirs operation (CHRO) optimization has attracted many researchers (Xu et al. 2013; Sharifi et al. 2021). The optimization of reservoir operation requires nearly no additional investment to increase generation capacity. CHRO is a complex nonlinear problem involved with multi variables. It needs to consider not only the hydraulic and electrical connections of the upstream and downstream reservoirs, but also many constraints in water level, reservoir release, power output, etc. (Sun et al. 2016; Chang et al. 2017; Gjorgiev & Sansavini 2018).

Over the past years, many attempts have been conducted to handle CHRO optimization problems (Kougias & Theodossiou 2013; Feng et al. 2020a, 2020b). Generally, methods used to solve CHRO problems can be roughly divided into conventional optimization algorithms like dynamic programming (Ji et al. 2014; Zhao et al. 2014), linear programming (Yoo 2009; Kang et al. 2018; Niu et al. 2018), nonlinear programming (Teegavarapu et al. 2013; Jothiprakash & Arunkumar 2014) and metaheuristic algorithms like genetic algorithm (GA) (Cheng et al. 2008; Haddad et al. 2016; Tayebiyan et al. 2016), partial swarm optimization (Kumar & Reddy 2007; Zhang et al. 2013, 2016) and differential evolution algorithm (Regulwar et al. 2010; Yazdi & Moridi 2018). Conventional optimization methods have been very successful in solving CHRO problems, but their performance is reduced in large and complex engineering problems. In dynamic programming, the high dimensionality of the problem poses difficulties and may not converge in a reasonable time, especially for large-scale hydropower systems (Zhang et al. 2014). Metaheuristic algorithms use a randomized optimization approach and do not rely on the gradient information of objective functions. In recent years, metaheuristic algorithms have gained substantial attention in reservoir scheduling due to their simplicity and ease of implementation (Yang et al. 2018; Dianatikhah et al. 2020). Hossain & El-Shafie (2014) compared GA, particle swarm optimization (PSO) and artificial bee colony (ABC) for reservoir operation optimization and found that the ABC algorithm exhibited the best performance. Mohammadrezapour et al. (2019) applied the cuckoo optimization algorithm (COA) for irrigation water management and optimized the total income of cultivation in the Qazvin plain. Ahmadianfar et al. (2022) proposed a multi-strategy slime mold algorithm to determine the optimal operating rules for hydropower multi-reservoir systems, and it showed good efficiency. Jiang et al. (2022) developed an elite collaborative search algorithm coupled with three improvement strategies and applied it to solve a joint hydropower generation dispatching model. Sharifi et al. (2022) investigated the capability of 14 recently-introduced robust evolutionary algorithms in the optimization of energy generation from the Karun-4 hydropower reservoir.

The butterfly optimization algorithm (BOA) is a nature-inspired metaheuristic algorithm recently developed by Arora & Singh (2018). In BOA, a set of solutions is randomly generated in the feasible space, and their positions are then iteratively updated according to global or local search equations. During the iteration, a switching probability constant is adopted to balance global exploration and local exploitation. BOA is simple in adaption for any optimization problem, easy to implement and has good optimization performance (Makhadmeh et al. 2022; Li et al. 2023). It has been widely applied in different domains in a short period, such as parameter identification (Wen & Cao 2020), feature selection (Arora & Anand 2019; Rodrigues et al. 2020; Sadeghian et al. 2021; Long et al. 2022), data classification (Tiwari & Chaturvedi 2022) and drought forecasting (Kisi et al. 2019). However, there are few reports of using BOA to address CHRO problems. To fill this gap, this paper attempts to verify the feasibility of BOA in solving reservoir operation optimization. Unfortunately, as with other swarm-based metaheuristic approaches, BOA tends to converge prematurely to a local optimum when applied to high-dimensional problems. Moreover, the movement of candidate solutions relies on a random selection of the local and global searches by a switching probability constant, which may lead to candidate solutions moving away from the global best solution (Tan et al. 2020). Therefore, further research is needed to enhance the balance of the local exploitation and global exploration ability in BOA and to improve the convergence speed. In this paper, an improved butterfly optimization algorithm (IBOA) was proposed based on three modified strategies: the self-adaptive strategy to improve the initial swarm, the dynamic switch strategy to enhance local search and levy-flight and standardized fragrance operators for position updating. In this study, the basic BOA and its improved version (IBOA) were used to solve a cascade reservoir operation problem. To test the performance, IBOA was compared to BOA and several famous optimization algorithms in solving numerical functions and cascade reservoirs’ optimal operation. More specifically, the main contributions were summarized as below: (1) An enhanced version of BOA was developed based on the self-adaptive strategy, the dynamic switch strategy and the Levy-flight and standardized fragrance operators. (2) The BOA and IBOA methods were compared to several popular algorithms in benchmark functions and a CHRO problem, including PSO, Sine Cosine algorithm (SCA), whale optimization algorithm (WOA) and tunicate swarm algorithm (TSA).

The goal of the model is to find the optimal water level of each reservoir at the end of each time period that maximizes the objective function without violating the following constraints.

• (1)
  Water balance constraint:

  

  (4)
• (2)
  Water storage constraint:

  

  (5)
• (3)
  Discharge constraint:

  

  (6)
• (4)
  Power generation constraint:

  

  (7)

  where and are the storage of the ith reservoir at the end of time period t and t + 1, respectively. ⁠, and are the inflow, power release and non-power release (spill) of the ith reservoir at the end of period t. and are the upper and lower bounds of storage for the ith reservoir at the end of period t. and are the upper and lower bounds of power release for the ith reservoir at the end of period t. and are the upper and lower bounds of power output for the ith reservoir at the end of period t.

The BOA is developed based on the foraging behavior of butterflies. Each butterfly yields a fragrance with a certain intensity, which is related to its fitness (objective function). For instance, when a butterfly moves from one position to another, its fitness changes consequently. The fragrance spreads through distance and other butterflies can detect it. This is the way in which butterflies share their personal information with others and create a collective social knowledge network (Arora & Singh 2018). When a butterfly can perceive a fragrance from the best butterfly in the solution space, it will move toward it and this phase is called a global search. In the alternative, when a butterfly cannot pick up a fragrance from its surroundings, it will move randomly and this phase is called local search.

The iteration stage proceeds until the predefined maximum number of iterations is reached. In the final stage, the BOA outputs the best solution found and its corresponding fitness value. The pseudo-code of the basic BOA can be found in Arora & Singh (2018).

Moreover, compared with the direct multiplication of fragrance in Equations (9) and (10), the use of a standardized fragrance operator allows setting different position updates according to the individual's fitness, eliminating the effect of the magnitude of fragrance on the position updates. If the fragrance of an individual's position f_i is close to the optimal value f_max, the variation of its position update is changed relatively small, and vice versa.

In BOA and IBOA methods, a fixed combination of parameters was adopted for both benchmark functions and cascade reservoir operation. The sensory modality c is 0.01 and the initial power exponent a is 0.1, and the initial switch probability is 0.8 as recommended by Arora & Singh (2018). Moreover, the results of several popular evolutionary algorithms are introduced for comparison, including PSO (Kennedy & Obaiahnahatti 1995), SCA (Mirjalili 2016), WOA (Mirjalili & Lewis 2016) and TSA (Kaur & Dhiman 2020). In the experiments on benchmark functions, the population and maximum iterations are set as 30 and 500. Referring to the parameter settings in the literature (Chang & Chang 2009; Ehteram et al. 2017), in the application to cascade reservoir operation, the population and maximum iterations are set as 1,000 and 1,000 in this study. According to the recommended empirical values from the literature (Feng et al. 2020a, 2020b; Kaur & Dhiman 2020), the other parameters are given below:

PSO: The inertia weight decreases from 0.9 to 0.1, while two learning factors are set as 2.0.

SCA: The parameter a₀ value is set as 2.0.

WOA: The parameter a₀ value decreases from 2 to 0.

TSA: The parameters p_min and p_max are set as 1 and 4, respectively.

To test the performance of the six mentioned algorithms in benchmark functions, 19 classical test functions are investigated (Digalakis & Margaritis 2001; Khishe & Mosavi 2020). These test functions roughly belong to two groups: unimodal (F₁–F₇) and multimodal functions (F₈–F₁₉), the specifications of which are given in Table 2. The variable denotes the dimension of the function (problem), f_min is the minimum value reported in the literature and range is the feasible space. Unimodal functions have only one global optimal solution, so they can test the exploitation and convergence speed of the algorithms. Conversely, multimodal functions have many locally optimal solutions, so they can test the exploration and local optima avoidance of the algorithms.

Table 2

Details of 19 benchmark functions

No. .	Function .	Variable .	Range .	fmin .
1		10	[−100,100]	0
2		10	[−10,10]	0
3		10	[−100,100]	0
4		10	[−100,100]	0
5		5	[−30,30]	0
6		10	[−100,100]	0
7		10	[−1.28,1.28]	0
8		10	[−5.12,5.12]	0
9		10	[−32,32]	0
10		10	[−600,600]	0
11		10	[−50,50]	0
12		10	[−50,50]	0
13		2	[−65.536,65.536]	1
14		4	[−5,5]	0.00030
15		2	[−5,5]	0.398
16		2	[−2,2]	3
17		4	[0,10]	− 10.1532
18		4	[0,10]	− 10.4028
19		4	[0,10]	− 10.5363

No. .	Function .	Variable .	Range .	fmin .
1		10	[−100,100]	0
2		10	[−10,10]	0
3		10	[−100,100]	0
4		10	[−100,100]	0
5		5	[−30,30]	0
6		10	[−100,100]	0
7		10	[−1.28,1.28]	0
8		10	[−5.12,5.12]	0
9		10	[−32,32]	0
10		10	[−600,600]	0
11		10	[−50,50]	0
12		10	[−50,50]	0
13		2	[−65.536,65.536]	1
14		4	[−5,5]	0.00030
15		2	[−5,5]	0.398
16		2	[−2,2]	3
17		4	[0,10]	− 10.1532
18		4	[0,10]	− 10.4028
19		4	[0,10]	− 10.5363

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Table 3 gives the statistics of six algorithms for the benchmark functions in 10 independent runs, including the average (Ave.) and standard deviation (Std.) of objective values. According to the average values, the PSO, BOA and SCA methods have comparable performance on the benchmark functions, both of which ranks 4–6 on 13 test functions, and ranks 1–3 on six test functions. The WOA method ranks 4–6 on six test functions, and ranks 1 on four test funcions. The TSA method ranks 4–6 on nine test functions, and ranks 1 on two test functions. It is obvious that IBOA has effectively outperformed some of the other techniques in most experimental functions. The IBOA method ranks 1 on 16 test functions, ranks 2 on two test functions and ranks 3 on one test function. It can be observed that in most functions (F₃–F₁₃, F₁₇–F₁₉). The WOA and TSA methods have better results on test functions than the PSO, BOA and SCA methods. For the ease benchmark functions, the IBOA method may not be the best, but it still yields a near-optimal solution. Furthermore, based on the Std. values reported in Table 3, it can be clearly shown that the suggested IBOA method has a higher level of reliability and accuracy compared to other optimization methods. IBOA achieves better results than BOA on 18 of the 19 tested functions in terms of Ave. and Std., indicating that the exploitation and exploration ability of BOA has been improved by the suggested three strategies. Overall, the IBOA method can produce satisfying global optimal solutions for 19 benchmark functions.

In this case, the operational aim is to maximize the total hydropower production of the cascade reservoir system in a year. Three typical inflow cases (wet year, normal year and dry year) are considered. Table 4 lists the statistics of hydropower output by six algorithms in 10 independent runs. The IBOA outperforms other algorithms in most statistical metrics of hydropower production. For instance, the worst solutions of IBOA are usually higher than the best solutions of other methods. Compared with PSO, BOA, SCA, WOA and TSA methods, the average hydropower production calculated by IBOA in the dry year is lifted by 0.5, 8.6, 4.7, 2 and 2%, respectively; whereas the improvements of the average production in the normal year are about 0.6, 11.2, 7.7, 1.9 and 0.6%, respectively. The IBOA method increases by 2.24, 18.97, 25.11, 1.43 and 0.57 hundred million kW h in contrary to the PSO, BOA, SCA, WOA and TSA methods, respectively. The lowest hydroelectric production was calculated using BOA and SCA methods for all runoff scenarios. In the dry year, the hydropower calculated by the PSO method is higher than the WOA and TSA methods. In the normal and wet years, the hydropower calculated by the TSA method is higher than the PSO and WOA methods. In addition, the standard deviation of IBOA is much smaller than those of the control methods. These statistical results demonstrate the search capability and convergence efficiency of IBOA are satisfactory when applied to cascade reservoir operation optimization problems.

Cascade reservoirs’ optimal operation aims to determine the best water release strategy for the hydroelectric system, which is a nonlinear, high dimension and restricted optimization problem. To address this issue efficiently, massive metaheuristic optimization methods have been explored. BOA is a novel metaheuristic optimization algorithm which has not been used in reservoir optimization problems. As stated by the No Free Lunch (NFL) theorem (Wolpert & Macready 1997), no single algorithm can yield a superior outcome for all optimization problems or even for instances of the same problem. Therefore, the basic form of any metaheuristic algorithm, such as BOA, can be modified or hybridized to align with the problem-solving requirements (Makhadmeh et al. 2022). In this paper, an improved butterfly optimization method (IBOA) is developed to solve the optimization problem of cascade hydropower reservoirs. Three effective strategies are used in IBOA: the self-adaptive strategy to improve initial population, the dynamic switch strategy to balance exploration and exploitation, Levy-flight and standardized fragrance operators for position updating. Basically, through the statistics and analysis of the optimization results, the following conclusions are obtained: (1) The IBOA method is verified and compared with several famous evolutionary methods (PSO, SCA, WOA BOA and TSA) on 19 numerical functions. IBOA's performance is the most impressive, achieving the best results in 16 out of 19 test functions compared to the rest algorithms. (2) The IBOA method is used to solve hydropower production optimization of cascade reservoirs. Results show that compared to other algorithms (PSO, SCA, WOA BOA and TSA), the IBOA method can produce better scheduling schemes with higher total hydropower output under different inflow cases.

Hence, the IBOA is an alternative way of addressing the cascade reservoir operation problem. In addition, the involved parameters in IBOA can be further analyzed and the multi-objective versions of BOA can be investigated to solve multi-objective reservoir optimization operation issues.

This work was supported by the National Natural Science Foundation of China (41730750), the Fundamental Research Funds for the Central Universities (B210203076), and China Scholarship Council (202006710113).

Zhangling Xiao and Zhongmin Liang conceptualized the whole article, developed the methodology, supervised the work, wrote the article, performed the investigation, conducted funding acquisition, and helped with programming. Jian Wang, Binquan Li, Yiming Hu and Jun Wang reviewed the article, conducted format analysis and rendered support in data curation.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.