## Abstract

For the low efficiency and large loss of cascade pumping stations, aiming to maximize system efficiency, an optimized scheduling model of cascade pumping stations is established with consideration of multiple constraints, and the optimal scheduling method based on the improved sparrow search algorithm (BSSA) is proposed. The BSSA is initialized by the Bernoulli chaotic map to solve the insufficient initial diversity of the sparrow search algorithm (SSA). The random boundary strategy is introduced to avoid local optimum when dealing with the scheduling problem of pumping stations. The performance and improvement strategy of BSSA are verified by eight benchmark functions. Results show that BSSA has better convergence accuracy and faster speed. BSSA is applied to a three-stage pumping station considering three flow conditions, and compared with the current scheme, particle swarm optimization and genetic algorithm optimization schemes, the operation efficiency of SSA can be increased by 0.72–0.96%, and operation cost can be reduced by ¥263,000/a–¥363,300/a. On this basis, the improvement of 0.04–0.30% and ¥14,800/a–¥109,900/a can be further achieved by the BSSA, which confirms the feasibility and effectiveness of BSSA to solve the pumping station optimal scheduling problem. The findings present useful reference for the optimized scheduling of pumping station system.

## HIGHLIGHTS

An optimal scheduling model of cascade pumping stations is established.

The sparrow search algorithm is improved as BSSA by using the Bernoulli chaotic map and random boundary treatment.

An optimization method for cascade pumping stations based on BSSA is provided.

The feasibility and effectiveness of the proposed method are confirmed.

## INTRODUCTION

The pumping station is an important facility in the fields of interbasin water transfer, residential water consumption, and drainage. Limited by the uneven distribution of water resources and the conditions on topography and geology, the joint operation of cascade pumping stations gradually has become the significant application form. However, due to the composition diversity of pumping stations and the complex hydraulic relationship among adjacent buildings and facilities, there often exist problems such as low overall operation efficiency and large energy consumption loss in the operation of cascade pumping stations. The advantages of optimal scheduling of pumping stations are the rational allocation of water resources, improving the utilization of water energy, as well as the reduction of system operation cost. Therefore, the optimal and scientific dispatch of cascade pumping stations has always been the focus of attention in the hydraulic engineering field.

Essentially, the optimized scheduling of cascade pumping stations includes a type of complex, nonconvex, nonlinear, and high-dimensional problem, with the core lying on the establishment of reasonable optimal scheduling model and the selection of appropriate solutions. Currently, traditional research methods mainly include linear programming (Shawwash *et al.* 2000; Sulis 2017), nonlinear programming (Guan *et al.* 1995), and dynamic programming (Delipetrev *et al.* 2015; Gong & Cheng 2018; Lei *et al.* 2018; Wen *et al.* 2022). Nonetheless, most of these methods have drawbacks such as unstable convergence results and long calculation time (Ming *et al.* 2015). The swarm intelligence algorithms such as particle swarm optimization algorithm (PSO) (Fu *et al.* 2011; Zhang *et al.* 2013, 2014; Mansouri *et al.* 2022), genetic algorithm (GA) (Moradi-Jalal & Karney 2008; Zheng *et al.* 2013; Yaghoubzadeh-Bavandpour *et al.* 2022; Zhuo *et al.* 2022), and cuckoo search algorithm (Meng *et al.* 2019; Ren *et al.* 2019; Trivedi & Shrivastava 2022) are broadly used in optimization problems about hydraulic engineering due to the advantages of simple principle, high searching accuracy, and strong optimizing ability. At the same time, intelligent algorithms can also be applied to predict various indicators in this field. Hadadi *et al.* (2022) used the adaptive neuro-fuzzy inference system and its hybrids with two bio-inspired optimization algorithms, namely, shuffled frog leaping algorithm and grey wolf optimization, in estimating monthly actual evapotranspiration in the Neishaboor watershed in Iran. The comparative analysis of the results showed that estimation errors of each part can be reduced, and higher prediction accuracy can be achieved by the three models. Mohammadi *et al.* (2022) and Mohammadi (2023) successively combined the algorithm with machine learning techniques and artificial neural networks. Then the algorithm was used to predict the rainfall runoff of the snow basin in Switzerland and the standardized precipitation index in Peru from 1990 to 2015. However, it should be pointed out that the conditions involving improper parameter setting, low population diversity, and so on may affect the performance of algorithms (Salgotra *et al.* 2023). Therefore, the development of new algorithms and continuous improvement of existing algorithms are sustaining. The constant need to seeking better algorithms and devotion to combining them with practical engineering have increasingly become a universal pursuit of the industry.

The sparrow search algorithm (SSA) is a swarm intelligent optimization algorithm proposed by Xue & Shen (2020), which is mainly inspired by the foraging and antipredation behavior of sparrow population. Since its invention in 2020, SSA has been investigated and applied in various fields due to its advantages such as fast convergence speed and good stability. For instance, through the proposal of an adaptive-tent chaos theory and theintroduction of the improved sine and cosine algorithm, the enhanced multistrategy sparrow search algorithm (EMSSA) was given by Ma *et al.* (2022), which improved the convergence speed and global exploration capability of the standard SSA, and the performance of EMSSA was verified comprehensively using 23 benchmark functions and CEC2014 and CEC2017 problems. Directing at the problem that traditional algorithms are susceptible to get into local optima and exhibit slow convergence in mobile robot path planning, an improved sparrow search algorithm (ISSA) based on the linear path strategy (LPS) was introduced by Zhang *et al.* (2022) in virtue of three strategies including LPS, new neighborhood search way, and new multi-index evaluation method. The results showed a shorter path and faster convergence through contrasting and comparing with different algorithms. Zhang & Han (2022) used a discrete SSA with a global perturbation strategy to solve the traveling salesman problem, achieving good convergence characteristics and robustness. Li *et al.* (2022a, 2022b) established the dynamic reconfiguration integrated optimization model of active distribution network and proposed a novel solving approach based on multi-objective SSA that effectively reduced the power loss and node voltage deviation. In addition, the SSA was also successfully put into use for parameter identification (Chen *et al.* 2021; Li *et al.* 2022a, 2022b), UAV flight path planning (Ouyang *et al.* 2021; Song *et al.* 2022; Yan *et al.* 2022; Yang *et al.* 2022), etc. It is worth noting that the SSA has not yet been applied to the field of cascade pumping stations, while the high convergence and calculation accuracy of the algorithm possess great potential in averting the optimal scheduling of cascade pumping stations from falling into local optimum with ease and improving operation efficiency.

Therefore, the SSA is introduced into the research on optimal scheduling of cascade pumping stations, and the corresponding optimization method based on BSSA is proposed to optimize the problem of local optimum at the boundary for the pumping station system, which is mainly induced by low initial population diversity and individual boundary aggregation. Taking the optimal scheduling of a three-stage pumping station as an example, the effectiveness and feasibility of the optimization scenario from BSSA are proved by comparing with the current scheme, the optimization scheme based on PSO, GA, and SSA under different flow conditions, so as to further ameliorate the total operation efficiency of cascade pumping stations and reduce the operation cost of the system.

## MATERIALS AND METHODS

### Optimal scheduling model of cascade pumping stations

The operation efficiency of the water conveyance system in cascade pumping stations is both the crucial index to reflect the running status of the system and the main factor to influence the cost of water conveyance. To elevate the operating efficiency of system, the pumping station is taken as the primary research target, the other buildings are generalized, and then the optimal scheduling model of cascade pumping stations can be established, taking the multifarious hydraulic constraints into account, accompanied by the achievement to maximize the system efficiency of cascade pumping stations.

#### Flow optimization distribution model of single-stage pumping station

The optimal flow distribution of the single pumping station can be treated as the optimization process to maximize the efficiency of the pumping station, which is carried out by determining the number of units and the flow allocation among units, under the condition that water transfer flow and working head are certain and on the premise of satisfying the constraints about flow balance and pump operation performance. The essence mentioned earlier is a spatial optimization problem referring to the flow distribution among pumping station units.

##### Objective function

*Q*

_{total}is the total flow rate of the pumping station,

*H*is the head of the

_{j}*j*th pumping station,

*η*is the total efficiency of the

_{j}*j*th pumping station in the case of

*Q*

_{total}and

*H*,

_{j}*Q*and

_{k}*η*represent the flowrate and efficiency of the

_{k}*k*th single pump in the

*j*th pumping station, respectively, and

*n*is the total number of pumps in the

*j*th pumping station.

##### Constraints

#### Head optimization model of cascade pumping stations

The head optimization of cascade pumping stations is the process to optimize the head distribution among pumping stations for the sake of maximizing the total system efficiency, when the overall head and flow are determined by meeting the constraints of hydraulic balance, and water level of inlet and outlet pond for various pumping stations. In essence, it is a spatial optimization problem of total head distribution intercascade pumping stations.

##### Objective function

*η*

_{cascade}is the total efficiency of cascade pumping stations where the

*Q*

_{total}is the flowrate and

*H*

_{total}is the head;

*η*

_{j}_{,max}is the maximum total efficiency of the

*j*th pumping station where

*Q*

_{total}is the flowrate and

*H*is the head.

_{j}##### Constraints

- (1)
Real hydraulic balance constraint:

- (2)
- (3)
- (4)

*Z*

_{1}and

*Z'*are the water levels of the inlet pool for the first pumping station and that of the outlet pond for the last pumping station, respectively;

_{m}*h*

_{j,j}_{+1}is the hydraulic loss of channel between the

*j*th and (

*j*

*+*1)th pumping stations;

*H*

_{j}_{,min}and

*H*

_{j}_{,max}denote the minimal and maximal heads of the

*j*th pumping station, respectively;

*Z*

_{j}_{,min}and

*Z*

_{j}_{,max}indicate the minimum and maximum water levels of forebay for the

*j*th pumping station, respectively; and

*Z'*

_{j}_{,min}and

*Z'*

_{j}_{,max}represent the minimum and maximum water level of outlet pond for the

*j*th pumping station, respectively.

#### One-dimensional hydraulic model

The one-dimensional hydraulic model is widely used in hydraulic engineering fields such as reservoir operation and cascade pumping station system optimization. In this article, the one-dimensional hydrodynamic model is established by using Saint–Venant equations to obtain the generalization of some complex internal structures including pumping station, inverted siphon, and gradient section in practical engineering, and the generalized structure model is coupled with the Saint–Venant equations, while the Pressmann four-point implicit difference scheme is adopted to discretize the equations generated, and double-sweep method is employed for the solution obtained.

### Introduction of improved sparrow search algorithm

#### Standard sparrow search algorithm

There is an obvious division of labor in the foraging process of the sparrow population, in which some are responsible for providing food directions and areas, while the rest complete the process as followers. When the sparrows detect danger, they will send out an alarm signal, and the population immediately makes antipredation behavior. The sparrow population can be further divided into discoverer, joiner, and scouter. The discover is in charge of furnishing foraging directions for other individuals. The joiner follows the discover to forage. The scouter takes charge of surveillance and early warning. The discoverer, joiner, and scouter use their own rules to update the position.

*t*is the current number of generations;

*T*is the constant value possessing the maximum iterations;

*α*is the random number between 0 and 1;

*i*= 1, 2, … ,

*N*, denotes the sort number of sparrow;

*d*= 1, 2, … ,

*D*signifies the dimension of current solution space;

*q*represents a random number following normal distribution;

*L*is a 1 ×

*D*matrix whose element value all equals to 1; and

*R*∈ [0, 1] and

*ST*∈ [0.5, 1] mean the prewarning value and safety threshold, respectively. When

*R*<

*ST*, the population is in the safe status and can be searched on a large scale. On the contrary, if

*R*>

*ST*, the scouters have discovered the predator and emitted an alarm signal, and the population immediately takes measures to move to the secure area.

*X*(

_{p}*t*+ 1) indicates the optimal position of discover at the (

*t*+ 1)th iteration;

*X*(

_{p}*t*+ 1) denotes the global worst location at the

*t*th iteration;

*A*is the matrix of 1 ×

*D*, in which 1 or −1 are selected randomly to assign for each element; and

*A*

^{+}=

*A*

^{T}(

*AA*

^{T})

^{−1}. When

*i*>

*N*/2, the joiners fail to obtain food that they should stay away from the current location. If

*i*<

*N*/2, the joiners move to a better position with discovers.

*X*

_{best}(

*t*) is the global optimal position;

*β*is a random number controlling step size and obeys normal distribution with a mean value of 0 and a variance of 1;

*K*∈ [−1, 1] is a random number;

*ε*is a relatively small constant with the purpose of avoiding the denominator being zero;

*f*is the fitness value of the

_{i}*i*th sparrow individual; and

*f*and

_{g}*f*represent the present global optimal and worst fitness values, respectively. When

_{w}*f*>

_{i}*f*, the sparrow individual is at the edge of group being readily detected by predator;

_{g}*f*=

_{i}*f*means that the individual is located in the middle of the population. To reduce the probability of being arrested, it is necessary to keep away from the current position and approach other individuals.

_{g}The parameters of the SSA algorithm are mainly composed of iteration number, population number, safety threshold, and the proportion of discoverers and scouters. Since the values of above parameters are different in the calculation of test functions and the optimization of cascade pumping station scheduling, the specific values are described in section 2.2.3 and Section 3.2.2 respectively.

The scheduling model of cascade pumping stations used in this paper contains two submodels, which include two variables, namely, flow and water level. Therefore, the symbol *X* either represents the unit flow in the optimization process of a single pumping station or denotes the water level of forebay and the outlet pond for pumping stations during the course of cascade pumping station optimization.

#### Improved sparrow search algorithm

For the swarm intelligence optimization algorithm, the quality and diversity of the initial population possess an excellent impact on the optimization performance for algorithms. Therefore, Bernoulli chaotic mapping is used in place of random initialization in the standard SSA algorithm to improve the quality of the initial population, which plays a role in improving the convergence speed and the ability to search for the global optimal solution. In addition, similar to amounts of traditional meta-heuristic algorithms, boundary processing is usually done by putting individuals at the boundary. However, the treatment will inevitably lead to a large number of individuals gathering at the boundary and decline the population diversity. In addition, with referring to the cascade pumping station, the upper and lower boundaries of the individual, namely, the extreme value of channel operation, are the highest and lowest water levels of forebay and outlet pond for the pumping station, respectively. What is more, considering the observation error, operation error, system error, and external disturbance in actual operation, the strategy of turning the individuals beyond the border into the boundary, in all probability, brings about the channel overflow and leakage. Therefore the irrationality to place individuals on the boundary in the optimal scheduling of cascade pumping stations, based on this consideration, a random boundary processing strategy is adopted as a replacement.

##### Bernoulli chaotic map

Chaos is the universal phenomenon in the nonlinear system, which is proverbially applied in optimizing search problems in view of the randomness, ergodicity, and regularity (Liu *et al.* 2016). Hence, the Bernoulli chaotic map is employed for population initialization in BSSA.

##### Random boundary treatment

*ub*and

*lb*are the upper and lower bounds of the individual range, respectively.

*et al.*2021):

When the individual is beyond the boundary, it can be randomly assigned again in the feasible region through Equation (15). Compared with the standard SSA, the random boundary strategy is able to augment the population diversity and then to improve the global search capability of the algorithm.

#### Verification of the proposed BSSA

##### Comparison of accuracy and convergence among different algorithms

To testify the feasibility of the improved algorithm and the corresponding optimization ability, the SSA, classical PSO, as well as GA are chosen for the comparative test on eight benchmark functions, whose basic information is presented in Table 1. Among them, F1 and F2 are unimodal functions, F3–F5 are multimodal functions, and F6–F8 represent the fixed-dimensional functions.

Function . | Dim . | Range . | f_{min}
. |
---|---|---|---|

30 | [−100, 100] | 0 | |

30 | [−100, 100] | 0 | |

30 | [−5.12, 5.12] | 0 | |

30 | [−32, 32] | 0 | |

30 | [−600, 600] | 0 | |

2 | [−65, 65] | 1 | |

6 | [0,1] | −3.22 | |

4 | [0, 10] | −10.4028 |

Function . | Dim . | Range . | f_{min}
. |
---|---|---|---|

30 | [−100, 100] | 0 | |

30 | [−100, 100] | 0 | |

30 | [−5.12, 5.12] | 0 | |

30 | [−32, 32] | 0 | |

30 | [−600, 600] | 0 | |

2 | [−65, 65] | 1 | |

6 | [0,1] | −3.22 | |

4 | [0, 10] | −10.4028 |

The four algorithms are compared in Matlab R2022b. To avoid the contingency of optimization results, each benchmark function is run independently for 30 times, the population size for each algorithm is set as 30, and the maximum number of iterations is 500. The parameters of algorithms are given as follows: the security value of SSA and BSSA is set to 0.8, and the proportion of discoverers and scouters is 20%. The inertia coefficient of PSO is [0.9, 0.6], and the acceleration constant is [2.1, 2.1]. The mutation probability and crossover ratio in GA are set to 0.01 and 0.8, respectively. The mean value, standard deviation, and two-sample *t-*test are selected as evaluation indexes. For the two-sample *t*-test results, ‘1’ indicates that BSSA or SSA is significantly superior, ‘0’ demonstrates that there is no difference between them, while ‘ − 1’ signifies that BSSA or SSA is obviously underprivileged, as shown in Table 2, where the bold data represent the optimal value of the index for the corresponding function.

F . | Index . | BSSA . | SSA . | PSO . | GA . | |||
---|---|---|---|---|---|---|---|---|

F1 | Mean | 9.3090×10^{−67} | 0.0000 | 2.8441×10^{−2} | 4.3360×10 | |||

Std | 5.0107×10^{−66} | 0.0000 | 8.0028×10^{−3} | 1.5708×10 | ||||

SSA vs others (t-stat t-test) | −1.9466 × 10 | 1 | −1.5119×10 | 1 | ||||

BSSA vs others (t-stat t-test) | 1.0176 | 0 | −1.9466 × 10 | 1 | −1.5119 × 10 | 1 | ||

F2 | Mean | 4.1486×10^{−52} | 8.4671×10 ^{−309} | 3.3004 | 1.8089×10^{4} | |||

Std | 2.2341×10^{−51} | 0.0000 | 1.1156 | 3.9263×10^{3} | ||||

SSA vs others (t-stat t-test) | −1.6203 × 10 | 1 | −2.5234 × 10 | 1 | ||||

BSSA vs others (t-stat t-test) | 1.0171 | 0 | −1.6203 × 10 | 1 | −2.5234 × 10 | 1 | ||

F3 | Mean | 0.0000 | 0.0000 | 2.6505 × 10 | 1.0975 × 10 | |||

Std | 0.0000 | 0.0000 | 6.0568 | 2.2667 | ||||

SSA vs others (t-stat t-test) | −2.3969 × 10 | 1 | −2.6520 × 10 | 1 | ||||

BSSA vs others (t-stat t-test) | 0.0000 | 0 | −2.3969 × 10 | 1 | −2.6520 × 10 | 1 | ||

F4 | Mean | 8.8818×10 ^{−16} | 8.8818×10 ^{−16} | 2.9249 | 2.9422 | |||

Std | 0.0000 | 0.0000 | 5.5663×10^{−1} | 2.0464×10^{−1} | ||||

SSA vs others (t-stat t-test) | −2.8780 × 10 | 1 | −7.8749 × 10 | 1 | ||||

BSSA vs others (t-stat t-test) | 0.0000 | 0 | −2.8780 × 10 | 1 | −7.8749 × 10 | 1 | ||

F5 | Mean | 0.0000 | 0.0000 | 1.6704×10^{2} | 1.3761 | |||

Std | 0.0000 | 0.0000 | 1.7647 × 10 | 1.9643×10^{−1} | ||||

SSA vs others (t-stat t-test) | −5.1844 × 10 | 1 | −3.8371 × 10 | 1 | ||||

BSSA vs others (t-stat t-test) | 0.0000 | 0 | −5.1844 × 10 | 1 | −3.8371 × 10 | 1 | ||

F6 | Mean | 1.3235 | 3.3402 | 9.9800×10 ^{−1} | 1.5758 | |||

Std | 1.7529 | 4.2507 | 2.2132×10 ^{−11} | 1.5260 | ||||

SSA vs others (t-stat t-test) | 3.0180 | −1 | 2.1398 | −1 | ||||

BSSA vs others (t-stat t-test) | −2.4023 | 1 | 1.0171 | 0 | −5.9458×10^{−1} | 0 | ||

F7 | Mean | − 3.3101 | − 3.2427 | − 3.2530 | − 3.2719 | |||

Std | 3.5668×10 ^{−2} | 5.6047×10^{−2} | 6.5222×10^{−2} | 6.0069×10^{−2} | ||||

SSA vs others (t-stat t-test) | 6.5353×10^{−1} | 0 | 1.9445 | 0 | ||||

BSSA vs others (t-stat t-test) | −5.5546 | 1 | −4.2080 | 1 | −2.9954 | 1 | ||

F8 | Mean | − 1.0403 × 10 | − 1.0049 × 10 | − 8.1489 | − 5.2089 | |||

Std | 0.0000 | 1.3259 | 3.0405 | 3.0215 | ||||

SSA vs others (t-stat t-test) | −3.1369 | 1 | −8.0337 | 1 | ||||

BSSA vs others (t-stat t-test) | −1.4639 | 0 | −4.0605 | 1 | −9.4155 | 1 | ||

BEST f_{min} | 5/8 | 5/8 | 1/8 | 0/8 | ||||

sig better | BSSA vs others (SSA vs others) | 2/8 | 7/8(6/8) | 7/8(6/8) | ||||

sig equiv | BSSA vs others (SSA vs others) | 6/8 | 1/8(1/8) | 1/8(1/8) | ||||

sig worse | BSSA vs others (SSA vs others) | 0/8 | 0/8(1/8) | 0/8(1/8) |

F . | Index . | BSSA . | SSA . | PSO . | GA . | |||
---|---|---|---|---|---|---|---|---|

F1 | Mean | 9.3090×10^{−67} | 0.0000 | 2.8441×10^{−2} | 4.3360×10 | |||

Std | 5.0107×10^{−66} | 0.0000 | 8.0028×10^{−3} | 1.5708×10 | ||||

SSA vs others (t-stat t-test) | −1.9466 × 10 | 1 | −1.5119×10 | 1 | ||||

BSSA vs others (t-stat t-test) | 1.0176 | 0 | −1.9466 × 10 | 1 | −1.5119 × 10 | 1 | ||

F2 | Mean | 4.1486×10^{−52} | 8.4671×10 ^{−309} | 3.3004 | 1.8089×10^{4} | |||

Std | 2.2341×10^{−51} | 0.0000 | 1.1156 | 3.9263×10^{3} | ||||

SSA vs others (t-stat t-test) | −1.6203 × 10 | 1 | −2.5234 × 10 | 1 | ||||

BSSA vs others (t-stat t-test) | 1.0171 | 0 | −1.6203 × 10 | 1 | −2.5234 × 10 | 1 | ||

F3 | Mean | 0.0000 | 0.0000 | 2.6505 × 10 | 1.0975 × 10 | |||

Std | 0.0000 | 0.0000 | 6.0568 | 2.2667 | ||||

SSA vs others (t-stat t-test) | −2.3969 × 10 | 1 | −2.6520 × 10 | 1 | ||||

BSSA vs others (t-stat t-test) | 0.0000 | 0 | −2.3969 × 10 | 1 | −2.6520 × 10 | 1 | ||

F4 | Mean | 8.8818×10 ^{−16} | 8.8818×10 ^{−16} | 2.9249 | 2.9422 | |||

Std | 0.0000 | 0.0000 | 5.5663×10^{−1} | 2.0464×10^{−1} | ||||

SSA vs others (t-stat t-test) | −2.8780 × 10 | 1 | −7.8749 × 10 | 1 | ||||

BSSA vs others (t-stat t-test) | 0.0000 | 0 | −2.8780 × 10 | 1 | −7.8749 × 10 | 1 | ||

F5 | Mean | 0.0000 | 0.0000 | 1.6704×10^{2} | 1.3761 | |||

Std | 0.0000 | 0.0000 | 1.7647 × 10 | 1.9643×10^{−1} | ||||

SSA vs others (t-stat t-test) | −5.1844 × 10 | 1 | −3.8371 × 10 | 1 | ||||

BSSA vs others (t-stat t-test) | 0.0000 | 0 | −5.1844 × 10 | 1 | −3.8371 × 10 | 1 | ||

F6 | Mean | 1.3235 | 3.3402 | 9.9800×10 ^{−1} | 1.5758 | |||

Std | 1.7529 | 4.2507 | 2.2132×10 ^{−11} | 1.5260 | ||||

SSA vs others (t-stat t-test) | 3.0180 | −1 | 2.1398 | −1 | ||||

BSSA vs others (t-stat t-test) | −2.4023 | 1 | 1.0171 | 0 | −5.9458×10^{−1} | 0 | ||

F7 | Mean | − 3.3101 | − 3.2427 | − 3.2530 | − 3.2719 | |||

Std | 3.5668×10 ^{−2} | 5.6047×10^{−2} | 6.5222×10^{−2} | 6.0069×10^{−2} | ||||

SSA vs others (t-stat t-test) | 6.5353×10^{−1} | 0 | 1.9445 | 0 | ||||

BSSA vs others (t-stat t-test) | −5.5546 | 1 | −4.2080 | 1 | −2.9954 | 1 | ||

F8 | Mean | − 1.0403 × 10 | − 1.0049 × 10 | − 8.1489 | − 5.2089 | |||

Std | 0.0000 | 1.3259 | 3.0405 | 3.0215 | ||||

SSA vs others (t-stat t-test) | −3.1369 | 1 | −8.0337 | 1 | ||||

BSSA vs others (t-stat t-test) | −1.4639 | 0 | −4.0605 | 1 | −9.4155 | 1 | ||

BEST f_{min} | 5/8 | 5/8 | 1/8 | 0/8 | ||||

sig better | BSSA vs others (SSA vs others) | 2/8 | 7/8(6/8) | 7/8(6/8) | ||||

sig equiv | BSSA vs others (SSA vs others) | 6/8 | 1/8(1/8) | 1/8(1/8) | ||||

sig worse | BSSA vs others (SSA vs others) | 0/8 | 0/8(1/8) | 0/8(1/8) |

*Note*: The bold data represent the optimal value of the index for corresponding function.

Table 2 shows that the BSSA has a great domination in the comparison of fixed-dimensional functions. For F6, BSSA is suboptimum; however, compared with SSA, the conspicuous presentation exists in the field of mean value, standard deviation, and two-sample *t*-test. In the calculation of F7, the average value of multiple optimization and standard deviation for BSSA are better than other algorithms, which is also the most prominent about two-sample *t*-test data. With reference to F8, although the improvement of optimization performance for BSSA is limited, the corresponding standard deviation of multiple optimization is 0; therefore, the strong stability of BSSA can be confirmed. Moreover, through the optimization contrast between unimodal and multimodal functions, the BSSA and SSA perform equally, and both prevail over PSO and GA dramatically.

In summary, with regard to eight benchmark functions, compared with SSA, the optimization performance of BSSA is markedly boosted. Especially for F6–F8, the fixed dimension functions that are difficult to be optimized, BSSA presents obvious advantages over the other three algorithms in optimization performance, which can efficaciously evade falling into local optimum, together with high optimization accuracy and strong search ability. Hence, the feasibility and superiority of BSSA are proved.

##### Analysis of improvement strategy

To further validate the optimization effect of Bernoulli chaotic map and random boundary processing on SSA, the following methods are utilized to test as follows:

- (1)
Initialization of Bernoulli chaotic map

The feasible region is divided into five parts equally, and the theoretical average number of individuals in each subdomain is 6. The amount of individuals in each subdomain is severally 7, 2, 3, 8, and 10 through random initialization, as illustrated in Figure 3(a), while that obtained by Bernoulli chaos initialization is 8, 8, 4, 5, and 5, as shown in Figure 3(b). Through comparison, it is found that the distribution of the individual population is more uniform and close to the ideal state when Bernoulli chaos is utilized for initialization. The mean and standard deviation of the 30 data from the two schemes are severally calculated, and the corresponding mean values are 0.59 and 0.46, while the standard deviations are 0.32 and 0.29, respectively. The results further demonstrate that the Bernoulli chaotic map can more evenly distribute the initial population and enhance the population diversity.

- (2)
Verification of random boundary processing strategy

## RESULTS AND DISCUSSION

### The solving steps of cascade pumping station model based on BSSA

- (1)Initialize the population. For the head distribution model of m-stage cascade pumping stations, the decision variables embody the water levels of the inlet and outlet pools of pumping stations. The initialization position of each decision variable is obtained by the following formula:where
*i*∈ [0, N],*j*∈ [1, m],*N*is the population number, and*Z*_{(i,j)}and*Z*'_{(i,j)}are the water levels of the inlet and outlet pools, respectively, of the*j*th pumping station in the*i*th individual from the population, respectively.*B*_{(i,j)}is the Bernoulli map value corresponding to the individual position. - (2)
Constraint processing. Constraint processing consists of equality and inequality constraints, and the overall processing flow is shown in Figure 6. First, determine whether the inequality constraint is satisfied. Provided that it is satisfied, then identify whether the equality constraint is fulfilled. If the equation constraint is attained, the processing process is completed; in the event that is not satisfied, the corresponding equation and inequality constraint are processed until both are satisfied. The procedure in this example is as follows: for inequality (7)–(9), Equation (15) strategy is used for constraint processing, that is, random values are re-evaluated within the desirable range. The equality constraint is dealt with through the strategy proposed by Tian

*et al.*(2014). By calculating the difference between the two sides of Equation (6), and distributing the difference to the water level before treatment, it is judged whether the newly generated*Z*and_{j}*Z'*violate the boundary conditions. With the help of continuous difference allocation and judgment, the water level of the inlet and outlet pools finally can meet the equality and inequality requirements under certain accuracy, and the precision here is accurate to 4 decimal places._{j} - (3)
Substitute the water level of inlet and outlet pools, as well as the flow with constraint treatment in the

*j*th pumping station, into the flow optimization distribution model of single-stage pumping station, and then start on the interior optimization in the*j*th pumping station: ② Equality and inequality constraints processing. The constraint processing method of Equation (2) and inequality (3) is the same as step 2.

③ Calculate the efficiency

*η*_{j}(0) of the*j*th pumping station using Equation (1), and save the maximum efficiency*η*_{j}_{,max}and the corresponding flow distribution.④ Update the position of discoverer, joiner, and scouter in the sparrow population by Equations (10)–(12), deal with the constraint of population after position updating, and calculate the corresponding efficiency

*η*_{j}(*t*) with Equation (1).⑤ Compare

*η*_{j}(*t*− 1) with*η*_{j}(*t*), update the efficiency*η*(_{j}*t*) and flow distribution in the*j*th pumping station when iteration reaches*t*, and then update the optimal efficiency*η*_{j}_{,max}and corresponding flow distribution for the current system.⑥ Let

*t*=*t*+ 1, return to step 4. Stop iteration until the termination condition is satisfied, and output the optimal efficiency*η*_{j}_{,max}and the corresponding optimal flow distribution in the*j*th pumping station.

- (4)
Calculate the system efficiency

*η*_{cascade}(0) making use of Equation (4), and save the maximum system efficiency*η*_{cascade,max}and the matching water level distribution. - (5)
Update the location of sparrow population according to Equations (10)–(12), and conduct constraint processing in accordance with step 2) and step 3).

- (6)
Update the efficiency

*η*_{cascade}(*t*) and the corresponding water level distribution of the cascade pumping station at the iteration number*t*, and afterward, update the system maximum efficiency*η*_{cascade,max}and the corresponding water level distribution. - (7)
Return to step 5, quit iteration until the termination condition is met, and output the optimal efficiency

*η*_{cascade,max}and the corresponding optimal water level distribution of the cascade pumping station.

### Case study

#### Project overview

^{3}/s. The design water level of the forebay of the first pumping station is 48.6 m and that of the outlet pond in the third pumping station is 51.82 m, assuming that the two values are constant. The electricity price of the pumping station is 0.8478 ¥/kw.h, which is the weighted average of the electricity price for different periods in the city. Under the three flow conditions, the relevant data of the current scheme are presented in Table 3.

Operation flow (m^{3}/s)
. | System Operation efficiency (%) . | Daily operating cost (¥) . |
---|---|---|

19.4 | 34.09 | 36,575 |

19.8 | 34.74 | 36,637 |

20.0 | 35.06 | 36,663 |

Operation flow (m^{3}/s)
. | System Operation efficiency (%) . | Daily operating cost (¥) . |
---|---|---|

19.4 | 34.09 | 36,575 |

19.8 | 34.74 | 36,637 |

20.0 | 35.06 | 36,663 |

#### Results and analysis

^{3}/s, as well as the bar chart of efficiency and annual operating cost obtained using the aforementioned algorithm. Moreover, the improved efficiency and cost savings of each algorithm optimization scheme compared with the current scheme are also marked in the figure.

##### Accuracy comparison of optimization schemes for cascade pumping stations

The figure also shows that compared with the current scenario, the optimization schemes based on algorithms under different flow conditions present various degrees of improvement in terms of both efficiency and cost. With the flow of 19.4 m^{3}/s, the system efficiency of cascade pumping station optimized by GA, PSO, SSA, and BSSA is 34.12, 34.15, 35.05, and 35.09%, respectively, which are improved by 0.03, 0.06, 0.96, and 1.00% in comparison with the current scenario. For the annual operating cost, namely, the ratio of the product of effective output power, annual operation duration and unit electricity price, to the operating efficiency in the cascade pumping station, the results calculated through four algorithms are ¥13,340,028, ¥13,326,283, ¥12,986,419, and ¥12,971,614, respectively. In contrast with the current scheme, ¥9,706, ¥23,451, ¥363,315, and ¥378,120 are economized, respectively. When the flowrate is 19.8 m^{3}/s, in contrast with the current scheme, the optimization results acquired through the aforementioned four algorithms are increased by 0.13, 0.15, 0.90, and 0.95% in terms of the efficiency, and ¥52,306, ¥59,306, ¥339,253, and ¥356,664 are saved from the perspective of annual operating cost, respectively. As the flowrate increases to 20.0 m^{3}/s, the efficiency is increased by 0.18, 0.08, 0.90, and 1.20%, respectively, and accordingly, the annual operating cost has the thrift of ¥69,597, ¥28,744, ¥332,624, and ¥442,538.

According to Section 1, it can be seen that compared to the efficiency improvement within 0.2%, and the annual cost saving within ¥70,000 obtained in virtue of GA and PSO optimization schemes, the effect of efficiency increased by 0.9%, as well as the corresponding cost saving of ¥330,000 acquired by SSA is quite significant. What is noteworthy is that the optimization results through BSSA can further improve the aforementioned indicators by, at least, 0.04% and ¥14,000 on the basis of SSA. Therefore, it can be concluded that the optimal scheduling method of cascade pumping station based on BSSA possesses obvious advantages.

##### Convergence comparison of optimization schemes about SSA and BSSA

^{3}/s condition, where the flow can be evenly divided and easy to be optimized, the iteration accuracy of BSSA is higher than that of SSA, and the convergence is achieved at the 13th iteration, which is significantly ahead of the result obtained by SSA that the number of iteration equals to 38. For the flow conditions of 19.4 and 20.0 m

^{3}/s with unbalanced distribution and tough optimization, although the convergence speed of BSSA lags behind slightly compared with SSA, the performance of BSSA is still superior to SSA in terms of convergence accuracy.

It can be concluded from the aforementioned analysis that, in comparison with the current scenario, the optimization scheme based on each algorithm can improve the system operation efficiency to varying degrees, and among them, the SSA and BSSA present the superiority. However, compared with the amplification of 27, 20, and 22.2%, in the field of computational efficiency optimization of the stochastic configuration network model (Zhang & Ding 2021), microgrid cluster economic optimization (Wang *et al.* 2021), and the cost optimization of robot path planning (Ouyang *et al.* 2021), by using the SSA algorithm, respectively, the efficiency improvement range increased in the optimization scheduling of cascade pumping stations is 2.56–2.82%. Even for the BSSA algorithm, the efficiency promotion range is also relatively limited, ranging from 2.72 to 3.42%. The phenomena are ascribed to the followings: (1) The net head of the cascade pumping station is only 3.22 m, which belongs to the water diversion project of low head among multitudinous cascade pumping stations; meanwhile, the ratio of head loss to gross head is above 0.3, and thus, the holistic head optimization range is small. In addition, the type of units in each pumping station is the same, and the optimization space above the flow level is also limited. (2) In the operation of the cascade pumping station, on the one hand, it is necessary to coordinate the complex hydraulic relationship between various buildings and hydraulic facilities, and on the other hand, the flow distribution of units in the pumping stations and the head combination among pumping stations are also influenced by manifold factors, so the overall optimization is more arduous.

However, it needs to be emphasized that the cascade pumping stations have the diffused existence in modern industry and agriculture, while the whole proportion of low-lift ones is remarkable. Although the BSSA has limited optimization space for a single project, the results show that compared with PSO, GA, and SSA, the proposed algorithm in this article presents better performance in the optimal scheduling of the cascade pump station. Considering the huge scale of the industry served, even if the optimization range is relatively small, the economic benefits and social value brought by BSSA cannot be ignored.

## CONCLUSION

In this article, an ISSA is put forward by means of adopting the Bernoulli chaotic map to initialize and improve the initial population diversity. At the same time, the random boundary processing strategy is used to optimize the individual boundary aggregation phenomenon caused possibly by the default boundary processing. The benchmark functions are selected to verify the convergence speed and search accuracy of the algorithm. On this basis, an optimal scheduling method of cascade pumping station based on BSSA is proposed and applied to solve the optimal scheduling of a three-stage pumping station. The conclusions are as follows:

- (1)
The mean value and the standard deviation calculated by BSSA in eight benchmark functions including Sphere, Rastrigin, and Hartmann 6-D are superior to that obtained through PSO, GA, and SSA, with faster convergence, indicating that the BSSA has better convergence accuracy, stronger robustness, and higher convergence speed, which verifies the effectiveness of the improved strategy.

- (2)
The flow conditions of 19.4, 19.8, and 20.0 m

^{3}/s are taken as examples, compared with the current scenario, the optimization scheme based on BSSA can severally improve the system operation efficiency by 1.00, 0.95, and 1.20%, and the annual operation cost is saved by ¥378,120, ¥356,664 and ¥442,538, respectively. In addition, in contrast with GA and PSO, the results acquired by using SSA can also be improved to some extent.

In the future, the research on multi-objective optimization scheduling of cascade pumping stations will be carried out with different head and flow, considering energy consumption, operating cost, and so on, so as to provide a more comprehensive optimization scheme for the system operation of the cascade pumping station. It should be noted that the research results of this article are still in the prediction stage. If conditions permit in the future, the authors will implement the scheme into the actual operation of the cascade pumping station to obtain better results for verification.

## ACKNOWLEDGEMENTS

This work was supported by the Basic Research Programs of Shanxi Province (grant no. 202103021224086, 20210302124645, 202203021222112); the National Key R&D Program of China (grant no. 2021YFC3001000); the Open Research Fund of Henan Key Laboratory of Water Resources Conservation and Intensive Utilization in the Yellow River Basin (grant no. HAKF202104); and the National Natural Science Foundation of China – Youth Program (grant no. 52109087).

## DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.

## CONFLICT OF INTEREST

The authors declare there is no conflict.