Optimization Method for Digital Scheduling of Oilfield Sewage System

Abstract

Oilfield Sewage System Scheduling is a complicated, large-scale, nonlinear system problem with multiple variables. The complexity of the sewage system pipeline network connection grows along with the ongoing building of oilfield stations, and the shortcomings of the sewage system water quantity scheduling program based on human experience decision-making become increasingly apparent. The key to solving this problem is to realize the digital and intelligent scheduling of sewage systems. Taking the sewage system of an oil production plant in Daqing oilfield as the research object, the water scheduling model of the sewage system is established in this paper. Aiming at the complex nonlinear characteristics of the model, the Levy flight speed updating operator, the adaptive stochastic offset operator, and the Brownian motion selection optimization operator are established by taking advantage of the particle swarm optimization (PSO) and the cuckoo search (CS) algorithm. Based on these operators, a hybrid PSO-CS algorithm is proposed, which jumps out of the local optimum and has a strong global search capability. Comparing PSO-CS with other algorithms on the CEC2022 test set, it was found that the PSO-CS algorithm ranked first in all 12 test functions, proving the excellent solving performance of the PSO-CS algorithm. Finally, the PSO-CS is applied to solve a water scheduling model for the sewage system of an oil production plant in Daqing Oilfield. It is found that the scheduling plan optimized by PSO-CS has a 100% water supply rate to the downstream water injection station, and the total energy consumption of the scheduling plan on the same day is reduced from 879.95 × 10⁶ m⁵/d to 712.84 × 10⁶ m⁵/d, which is a 19% reduction in energy consumption. The number of water balance stations in the sewage station increased by 7, which effectively improved the water resource utilization rate of the sewage station.

1. Introduction

Oilfield sewage is a by-product of the oilfield extraction process, consisting of oilfield-produced water, drilling sewage, and other types of oily sewage in the sewage station [1]. It is the primary issue with the water quality in oilfields and presents a major risk to human health and the environment [2]. The sustainable growth and use of water resources are realized through the effective functioning of the oilfield sewage system, which also lowers environmental contamination and encourages clean oilfield production [3,4]. Though labor-intensive and time-consuming, the current oilfield sewage system water scheduling scheme primarily depends on human experience, which increases the likelihood of operational and technical issues. Moreover, it is challenging to guarantee the water scheduling scheme’s quality in real time [5,6]. Digital scheduling is based on the digitization of production demand, and dynamic water allocation is achieved daily through remote regulation. However, there is no scientific decision-making method for digital scheduling. Therefore, it is of great significance to carry out an optimization method for the digital scheduling of oilfield wastewater systems in order to improve the operational efficiency of oilfield wastewater systems, the utilization of water resources, and the protection of the environment.

Relatively few studies have been performed recently on the subject of oilfield sewage system water scheduling optimization. The research field of water resources scheduling includes the study of water scheduling in oilfield sewage systems, and many academics have worked extensively in this area to better understand how water resources are scheduled. Jia et al. [7] propose a multi-stage reservoir scheduling and water allocation model based on multi-stage coupling. The model can select more reasonable water transfer methods according to different development needs so that the water shortage rate of each user in the receiving area is kept below 8% and the reservoir idle rate is controlled at about 10%, which effectively improves the water resource allocation capacity. Yang et al. [8] establish a multi-objective optimal scheduling model and splits the multi-objective optimal scheduling model using a two-stage method in order to optimize the water transfer engineering scheme. Wu et al. [9]. proposed a multi-objective reservoir optimal scheduling model as a water transfer rule for inter-basin water transfer with the objectives of economic efficiency, power generation, and ecological water transfer. Wang et al. [10] developed a mixed-integer linear programming model considering water demand, pumping station transmission capacity, and pipeline operation constraints. The model minimizes the total cost of the oilfield water delivery system and improves water resource utilization. However, with the increasing complexity of the water resources scheduling operation model, traditional algorithms such as linear programming, nonlinear programming, and dynamic programming can no longer meet the development trend of the water allocation model [11].

Heuristic optimization methods, which may quickly converge to the global optimal solution in a complicated solution space, have become widely employed in water resource allocation research due to advancements in computer technology and artificial intelligence [12]. Under difficult restrictions, it offers an effective solution approach for the water resource allocation model [13]. Ezzeldin et al. [14] proposed a novel dual hybrid optimization algorithm model of Genetic Algorithm-Particle Swarm Algorithm to solve the problem of optimal design of urban water distribution networks to support urban water allocation. Deng et al. [15] proposed a deoxyribonucleic acid (DNA) algorithm as a biological algorithm for solving the rural postman’s problem, which solves the water allocation problem. Turci et al. [16] proposed an improved genetic algorithm to solve the pump scheduling model. Dong et al. [17] proposed a data-driven water pumping station unit optimization scheduling model based on the fusion of IoT data and PSO using a Long Short-Term Memory (LSTM) network for optimal scheduling of water pumping stations using a predictive control framework. The particle swarm algorithm and its improvement methods have become one of the most widely used optimization algorithms in water scheduling engineering due to its advantages, such as easy implementation, algorithmic simplicity, and high efficiency [13]. Eryiğit et al. [18] proposed a modified immune algorithm to solve an urban water supply scheduling model in the Iraqi desert for optimal allocation of urban water management. Ma et al. [19] developed a parallel particle swarm algorithm to solve a multi-objective scheduling model of a reservoir cluster in the Heihe River Basin. An adaptive stochastic inertia-weighted ARIW-PSO was proposed by Chen et al. [13] to address a multi-objective reservoir cluster scheduling model. PSO-GA was proposed by Wu et al. [20] as a solution to the Drinking River Basin multi-objective ecological dispatching model.

In this paper, an improved particle swarm optimization algorithm with a hybrid particle swarm-cuckoo (PSO-CS) algorithm is proposed in combination with a cuckoo intelligent optimization algorithm. The algorithm is used to compensate for the shortcomings of the standard PSO algorithm, which is easy to fall into the local optimum and has fast convergence speed. In order to accurately assess the reliability of the improved PSO, two non-parametric statistical tests were used: the Friedman test and the Bonferroni–Dunn test. The Friedman test is suitable for dealing with repeated measurements and is able to provide robust statistical results by effectively comparing the differences between multiple samples of interest without relying on the assumption of normal distribution. The Bonferroni–Dunn test, on the other hand, enhances the accuracy and reliability of the results when making multiple comparisons by adjusting the significance level to control the rate of false positives and reduce the risk of false findings. The combined use of these two tests ensures that our assessment of the performance of the improved particle swarm algorithm is more comprehensive and rigorous. Taking the production and operation situation of a sewage plant in Daqing oilfield as a case study, considering the oilfield sewage system process flow and water transfer rules, constructing a sewage system water quantity scheduling optimization model, and solving the sewage system water quantity scheduling model by using the improved PSO-CS algorithm. This study innovatively combines an improved particle swarm algorithm with a water scheduling model for oilfield wastewater systems. The optimization efficiency of the scheduling scheme is improved by introducing a swarm intelligent optimization algorithm. Compared with the traditional method, we used the improved particle swarm algorithm to enhance its adaptability and performance in complex wastewater treatment systems. The objective of this study is to provide an optimal water transfer scheme for the production and operation of oilfield wastewater systems through the improved particle swarm algorithm.

2. Materials and Methods

2.1. Overview of the Sewage System in the Study Area

This paper uses the Daqing oilfield in China as its study area. The study area’s sewage system is primarily made up of water-driven sewage treatment stations, poly-containing and ternary sewage treatment stations, deep sewage treatment stations, water injection stations, and a network of sewage pipelines. It consists of eleven water-driven sewage treatment plants, seven comprising poly and ternary treatment plants, nine depth treatment plants, and twenty-two water injection plants. To maintain the regular operation of the water injection system, the upstream sewage treatment station uses pipelines to deliver treated sewage containing oil to the downstream water injection station. Figure 1 depicts the sewage system’s overall layout. Water-driven, poly, and ternary sewage treatment stations with depth and water-driven capabilities to handle 5 mg/L of water quality and 20 mg/L of water quality, respectively. Using the plant’s daily sewage system production statistics from 2024 as an example, the system received 270,698 m³ of raw water each day, of which 164,964 m³ had a high concentration of poly sewage (carrying more than 150 mg/L of polymer). 105,734 m³ of low-containing poly sewage (with a poly content of less than 150 mg/L). The system discharged 109,646.3 m³, of which 63,264.95 m³ were ordinary water and 46,381.35 m³ were water at a depth. The scheduling scheme for this sewage system currently relies on manual experience, which makes it challenging to guarantee the system’s stable and efficient operation. It also faces issues with declining well production, the oilfield sewage system’s high energy consumption, high production costs, etc. Researching the improvement of the sewage system’s operating plan is necessary to boost the system’s operational efficiency and attain steady, uninterrupted oilfield operations.

2.2. Mathematical Model for Optimization of Water Dispatch in Sewage System

2.2.1. Objective Function

The goal of the oilfield sewage system schedule optimization is to identify the best scheme out of those available to satisfy the various needs of the system. The energy consumption of the system during the transfer process is used to weigh the benefits and drawbacks of each scheme. This paper uses the pipeline length, the product of the pipeline’s flow rate, and its pressure drop along the route to describe the energy consumption of the system, and this simplifies the model. The sewage system’s scheduling optimization process objective function may be stated as Formula (1).

$$\min F(Q)={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{Q}_{i,j}\times L_{i,j}}}\times \Delta H_{i,j}$$

(1)

$$\Delta H_{i,j}=\beta {Q_{i,j}}^{2-\mathrm{k}}{L}_{i,j}{V}^k/{D_{i,j}}^{5-k}$$

(2)

In Formulas (1) and (2),

$Q_{i,j}$—Volume of water in the pipeline from the sewage station $i$ to the water injection station $j$ in the system, ${\mathrm{m}}^3/\mathrm{d}$;

$L_{i,j}$—Length of pipeline from sewage station $i$ to water injection station $j$ in the system, $\mathrm{m}$;

$\Delta H_{i,j}$—Pressure drop along the pipeline from the sewage station $i$ to the water injection station $j$ in the system, $\mathrm{m}$;

$D_{i,j}$—Pipeline diameter from sewage station $i$ to water injection station $j$ in the system, $\mathrm{m}$;

$V$—Kinematic viscosity of the fluid in the pipe, ${\mathrm{m}}^2/\mathrm{s}$;

$m$—Number of sewer stations in the system, se;

$n$—Number of water injection stations in the system, se;

$\beta$—Pressure drop calculation factor;

$k$—Fluid state index, values are shown in

Table 1. Fluid state index value.

Flow Pattern		$\mathit{k}$
Laminar flow		1
Turbulent flow	Hydraulically smooth	0.25
	Mixed friction	0.123
	Completely rough	0

;

2.2.2. Constraint Conditions

The restrictions between the transfer pipeline network are established starting with the injection and sewage systems combined. Determining the volume of water to be moved from each sewage treatment station reservoir to the downstream injection station within the intricate network structure is the goal of the sewage system dispatch optimization strategy. Combining the needs and operational features of the water injection system and the sewage system results in the following constraints.

• Supply and demand constraints

The sum of the water volume of the upstream sewage treatment station of the oilfield sewage system is greater than or equal to the sum of the water required by each downstream water injection station.

$${\displaystyle \sum _{i=1}^m{a}_i}\ge {\displaystyle \sum _{j=1}^n{b}_j}$$

(3)

In Formula (3),

$a_i$—Volume of water from sewage station $i$, ${\mathrm{m}}^3/\mathrm{d}$;

$b_j$—Water requirements for water injection station $j$, ${\mathrm{m}}^3/\mathrm{d}$;

2.

Water quantity constraints

The total water output of any sewage station is equal to the sum of the water flow out of the station, which is equal to the sum of the flow in the pipeline in the direction of its downstream conveyance water, and the total water demand of any water injection station is equal to the sum of the water flow into the station, which is equal to the sum of the flow in the pipeline in the direction of its upstream incoming water.

$${\displaystyle \sum _{j=1}^n{Q}_{i,j}}=a_i,i=1,2,3,...,m$$

(4)

$${\displaystyle \sum _{j=1}^m{Q}_{i,j}}=b_i,j=1,2,3,...,n$$

(5)

$$Q_{i,j}\ge 0,i=1,2,3,...,m,j=1,2,3,...,n$$

(6)

In Formulas (4)–(6),

$Q_{i,j}$—Volume of water in the pipeline from the sewage station $i$ to the water injection station $j$ in the system, ${\mathrm{m}}^3/\mathrm{d}$;

$a_i$—Volume of water from sewage station $i$, ${\mathrm{m}}^3/\mathrm{d}$;

$b_j$—Water requirements for water injection station $j$, ${\mathrm{m}}^3/\mathrm{d}$;

$m$—Number of sewage stations in the system, se;

$n$—Number of water injection stations in the system, se;

3.

Injection station effluent concentration constraints

One of the indications used to assess whether the water transfer plan satisfies production demand is the poly concentration of sewage, which is a crucial parameter in the sewage system dispatch optimization process. Each downstream injection station in the sewage system must have incoming sewage concentrations that are either less than or equal to the injection concentration requirements of that station.

$${\displaystyle \frac{{\displaystyle \sum _{i=1}^m{Q}_{i,j}}\times {\eta }_i}{{\displaystyle \sum _{i=1}^m{Q}_{i,j}}}}\le {\eta }_{j,\max },j=1,2,3,...,n$$

(7)

In the Formula (7):

${\eta }_i$—Concentration at sewage station $i$, $\mathrm{mg}/\mathrm{L}$;

${\eta }_{j,\max }$—Maximum effluent concentration required at the jth injection station, $\mathrm{mg}/\mathrm{L}$;

$Q_{i,j}$—Pipeline flow from sewage station $i$ to water injection station $j$, ${\mathrm{m}}^3/\mathrm{d}$;

$m$—Number of sewage stations in the system, se;

$n$—Number of water injection stations in the system, se;

4.

Water injection station pressure constraints

The inlet pressure of each pipe at the injection station downstream of the sewage system is greater than or equal to the required pressure at the injection station.

$${P_j}^{in}\ge {P_j}^n$$

(8)

In Formula (8),

${P_j}^{in}$—Inlet pressure at the injection station $j$ in the system, Mpa;

${P_j}^n$—Required injection pressure at the $j$th injection station in the system, Mpa.

2.3. PSO-CS Algorithm

2.3.1. PSO Algorithm

PSO algorithm is a method to optimize continuous nonlinear function based on iterative updating of random population, which originates from the study of bird predation behavior [21]. The main process is that, in each iteration cycle, the potential position of each particle is randomly distributed between the individual historical optimal position and the population historical optimal position, and PSO uses the velocity–position update formula to make each candidate solution follow the current optimal solution, and ultimately find the global optimal solution after many iterations [22]. The mathematical model of the standard PSO is as follows.

The symbols $x$ and $v$ represent the position of the particle and the search velocity of the particle, which correspond to the solution of the problem and the search velocity of the solution, respectively. Suppose that in a D-dimensional target search space, there are N particles forming a population, the solution set is $X={(x_{1,}{x}_2,\cdot \cdot \cdot ,x_N)}^T$, where $x_i=(x_{i1},x_{i2},\cdot \cdot \cdot ,x_{iD})$ is the i-th solution and $v_i=(v_{i1},v_{i2},\cdot \cdot \cdot ,v_{iD})$ is the search velocity of the current solution. $pbes{t}_i=(pbes{t}_{i1},pbes{t}_{i2},\cdot \cdot \cdot ,pbes{t}_{iD})$ and $gbes{t}_i=(gbes{t}_{i1},gbes{t}_{i2},\cdot \cdot \cdot ,gbes{t}_{iD})$ are individual historical optimal solutions and global historical optimal solutions, respectively. The velocity iteration update formula is based on Equation (9), and the solution iteration update formula is based on Equation (10).

$${v_i}^{(t+1)}={\omega }^{(t)}\times {v_i}^{(t)}+c_1\times {r_1}^{(t)}(pbes{t_i}^{(t)}-{x_i}^{(t)})+c_2\times {r_2}^{(t)}\times (gbes{t_i}^{(t)}-{x_i}^{(t)})$$

(9)

$${x_i}^{(t+1)}={x_i}^{(t)}+{v_i}^{(t+1)}$$

(10)

where, ${\omega }^{(t)}$ is the inertia coefficient of the solution $x_i$, and the value ranges from 0.4 to 0.9 [23]. $c_1$ and $c_2$ are individual learning factors and social learning factors respectively, and both of which are positive numbers.

${r_1}^{(t)}$ and ${r_2}^{(t)}$ are uniform random numbers between 0 and 1.

The steps of the PSO optimization algorithm are described as follows:

Step 1: Randomly initialize the position ${x_i}^{(0)}$ and velocity ${v_i}^{(0)}$ (water flow rate of each pipeline from the sewage station to the injection station) of each particle, calculate the fitness value of ${x_i}^{(0)}$ (the objective function value of the mathematical model for optimization of the sewage system scheduling), and determine the values of $pbes{t}_i$ and $gbes{t}_i$.

Step 2: Update the positions and velocities of the particles according to Formulas (9) and (10), and then calculate the water dispatch energy consumption according to Formulas (1) and (2).

Step 3: Evaluate the fitness value of ${x_i}^{(t+1)}$. If the value is desirable (smaller value of the sewage system scheduling optimization objective function), then replace $pbes{t_i}^{(t)}$ and $gbes{t_i}^{(t)}$ with the new $pbes{t_i}^{(t+1)}$ and $gbes{t_i}^{(t+1)}$.

Step 4: If $t<T$, then $t=t+1$ and return to step 2; otherwise, optimization stops.

2.3.2. CS Algorithm

The cuckoo search algorithm (CS) is a meta-heuristic algorithm proposed to mimic the natural phenomenon of cuckoos laying eggs on borrowed nests. In the classical CS algorithm, the cuckoo nest location update is based on Levi’s flights [24]. In order to facilitate the simulation of the cuckoo’s reproduction strategy, the cuckoo algorithm will be assumed to be in the following three ideal states [25].

(1)

Each cuckoo lays one egg at a time and randomly selects a location for the nest to incubate;

(2)

In a randomly selected group of nests, the high-quality nests will be retained for the next generation;

(3)

The number of available nests is fixed, and the probability that the nest owner discovers an alien cuckoo egg is Pa ∈ [0, 1]. When a nest host discovers an alien cuckoo egg, it will either discard the cuckoo egg or re-establish a new nest.

Based on the above three rational states, the updated formula for the cuckoo’s location and path to find the host nest is as follows.

$$x_i(t+1)=x_i(t)+\alpha Levy(\beta )$$

(11)

In Formula (11), $\alpha$ is the step control factor, when $\alpha >0$, then the typical value $\alpha =1$; $Levy(\beta )$ denotes the Lévy random search path, where $\beta$ is the control parameter of the path, and the specific formula for $Levy(\beta )$ is shown below.

$$Levy(\beta )={\displaystyle \frac{\varphi u}{{\left|v\right|}^{\frac{1}{\beta }}}}$$

(12)

In the formula, $\varphi ={\left\{{\displaystyle \frac{\Gamma (1+\beta )\times \sin (\pi \times \beta /2)}{\Gamma \left\{[(1+\beta )/2]\times \beta \times 2^{(\beta -1)/2}\right\}}}\right\}}^{1/\beta }$; $u,v$ is a random number obeying a normal distribution.

The number of available nests is fixed, and the probability that the nest owner discovers an alien bird’s egg is Pa ∈ [0, 1]. After updating the location, a random number r is compared with Pa. If r > pa, the nest location is updated randomly once; otherwise, the nest location remains unchanged. The updated formula for the location of the discovered nest is as follows.

$$x_i(t+1)=x_i(t)+r(x_j(t)-x_k(t))$$

(13)

In Formula (13), $r$ is the scaling factor; $x_j(t),x_k(t)$ is the random position of the nest in the $t$th generation.

2.3.3. PSO-CS Hybrid Algorithm

1.

Feasibility Analysis

The optimization of individual search behavior is the key to improving the PSO-CS intelligent optimization algorithm. To demonstrate intuitively that the optimization performance will be effectively enhanced by mixing the two algorithms, Figure 2 is plotted to discuss the optimization mechanism of the mixing of the two algorithms. Particle $i$ moves to point 2 for the traditional PSO algorithm, as illustrated in Figure 2. The particle is then guided to point 3 by the historical optimal position and moves to point 4 from point 3 under the combined influence of social cognition and the current optimal position. That is, if no improvement occurs, the particle can move from point 1 to point 4. A large step movement can be performed by the individual with a certain probability when they move to point 1 of the PSO-CS algorithm. This step movement incorporates the Levy flight’s step movement and is drawn to the historically optimal position to point 5. Then, the individual performs an adaptive stochastic offset operator, taking into account the effect of the current group densities, using the offset operation to move point 6 from point 5. Ultimately, the PSO-CS algorithm’s addition of the Brownian motion selection operators causes the current optimal individual to perform a random Brownian selection operation within a specific neighborhood, effectively enhancing both the optimal position and the current optimal individual. As a result, the individual moves from point 6 to point 7. That means after adding the three optimization operators, the individual moves from point 1 to point 7. In comparison to the traditional PSO algorithm, the hybrid algorithm more effectively incorporates the CS algorithm’s stochastic search capability. Additionally, it increases the individual’s global optimization searchability and the probability of jumping out of the local optimal solution under the influence of the random offset and the current optimal individual selection optimization. This can also cause the algorithm to jump out of the local optimum, as demonstrated by the hybrid algorithm’s individual jumping out of the local optimal region (represented by the blue region in the figure). The aforementioned study leads to the conclusion that combining the PSO and CS algorithms to create a superior algorithm works well.

2.

Levy Flight Speed Update Operators

Levy flight generates a class of random motion trajectories distinct from Brownian motion and is a random walk with a heavy-tailed probability distribution of step lengths. Its movement pattern matches the pattern of birds that forage in one place for food but travel to other regions when food becomes scarce. It also moves with a high step length with a given probability. Levy flight is incorporated into particle updating based on the PSO algorithm’s process architecture. This results in the formation of a new Levy flight velocity updating operator by converting the conventional constant learning factor into Levy flight coefficients.

For the PSO algorithm, the speed update of particles is mainly performed concerning the historical optimal individuals and the current global optimal individuals. Although it accelerates the speed of the population to discover high-quality solutions to a certain extent, it causes the population to gather around excellent individuals and fall into the local optimum. To overcome the shortcomings of the PSO algorithm that the diversity decreases in the late iteration and makes the individuals jump out of the local optimum, combined with the advantage of Levy flight that can search for large step size, the following formula for the speed update of the individuals in the hybrid algorithm is obtained.

$$v_i(t+1)=w\cdot v_i(t)+L_v\cdot r_1\cdot [pbes{t}_i(t)-x_i(t)]+c_2\cdot r_2\cdot [gbest(t)-x_i(t)]$$

(14)

In this formula, $L_v$ is the Levy flight coefficient, which is calculated as follows:

$$L_v={\displaystyle \frac{\mu p{\left\{\frac{\Gamma (1+\beta )\times \sin (\pi \times \beta /2)}{\Gamma \left\{[(1+\beta )/2]\times \beta \times 2^{(\beta -1)/2}\right\}}\right\}}^{1/\beta }}{{\left|q\right|}^{\frac{1}{\beta }}}}$$

(15)

In Formula (15), $\beta$ is the control parameter of the Levy flight path, $\mu$ is the Levy flight control factor, and $p,q$ is a random number obeying a normal distribution.

3.

Adaptive Stochastic Offset Operators

By using the concept of randomly discarding individuals from the CS algorithm, the adaptive random offset operator selects a certain number of individuals from the population—excluding the currently optimal individuals—for random offset after each generation of the hybrid algorithm. This process can significantly increase the population’s capacity to explore all possible domains. The selection of these individuals is not according to the level of fitness value but according to the random selection. A different offset group can be created by randomly choosing a specific number of people. The alternative offset group can then use the change rule of group diversity to adjust the corresponding offset probability and assess whether or not to perform the offset operation. The effect of group density is taken into account when calculating the probability of offset individuals by measuring the size of the group diversity. For each individual in the alternative group, it is determined one by one if their offset probability crosses the threshold; if it does, the offset operation is carried out, as indicated by the discriminant formula of the offset operation of the adaptive optimization process.

$$P_A\left(x_i(t)\right)=e^{-\kappa /\min \left(x_j(t)-x_k(t)\right)}>{\lambda }_A,\min \left(x_j(t)-x_k(t)\right)\ne 0$$

(16)

In this formula, $P_A()$ is the probability of the $i$th individual’s bias; $\kappa$ is the bias control parameter; $\min ()$ denotes the minimum value of the formula taken over the inter-vectorial formula of individuals in parentheses; $x_j(t),x_k(t)$ is the randomly selected individual in the population of the hybrid algorithm; and ${\lambda }_A$ is the bias threshold.

For the individuals whose offset probability is greater than the threshold, the offset is performed based on the following formula to obtain new individuals, and the new individuals are mixed into the original population to continue performing the subsequent iterations.

$${x^{\prime }}_i(t)=x_i(t)+r_A(t)\left(x_j(t)-x_k(t)\right)$$

(17)

In the formula above, $x_i(t)$ is the vector form of the individual performing the offset operation through the offset discrimination; ${x^{\prime }}_i(t)$ is the vector form of the individual after performing the offset operator; $x_j(t),x_k(t)$ is the vector of the two randomly selected individuals in the discrimination formula; and $r_A(t)$ is the scaling factor, which is calculated as shown in the following Formula (18):

$$r_A(t)=r_{\min }+(r_{\max }-r_{\min }){\displaystyle \frac{\pi t}{2I_{\max }}}$$

(18)

In this formula, $r_A(t)$ is the scaling factor of the $t$th iteration, $r_{\min }$ is the minimum value of the scaled factor, $r_{\max }$ is the maximum value of the scaling factor, and $I_{\max }$ is the maximum number of iterations.

4.

Brownian Motion Selection Operators

Brownian motion random search is used to optimize the entire population after improving the current optimal individual, hence expanding the algorithm’s global search capability. The selection space of the current optimal individual is generated by the Brownian motion using a random step. Each state in the space is a possible state of the current optimal individual, and the selection space is denoted by the $\{g{b}^1,g{b}^2,\cdots ,g{b}^{mB}\}$. The current optimal individual and the step of the Brownian motion combine to form each state in the selection space, with $g{b}^j$ representing the $j$th state and $m_B$ denoting the number of states. Each state in the selection space consists of a combination of the current optimal individual and the Brownian motion step, which is formulated as follows:

$$g{b}^j=gb(t)+\chi B_s(t)$$

(19)

In this formula, $gb(t)$ is the current optimal individual of the $t$th iteration, $\chi$ is the Brownian motion step size control factor, and $B_s$ is the approximate Brownian motion step size, where each dimension of the step size is calculated as shown in the following formula:

$$B_{s,j}(t)={\displaystyle \sum _{k=0}^{N}ak\frac{\sqrt{2}}{k+\frac{1}{2}}\sin \left((k+\frac{1}{2})\pi t\right)}$$

(20)

$B_{s,j}(t)$ is the value of the $j$th dimension of the approximate Brownian motion step; $a$ is a random number from a standardized nontrivial distribution; and $N$ is a positive integer corresponding to the Brownian motion.

Based on the above selection space, the best state adapted by the current optimal individual is selected from this space, and the specific formula for the optimal selection has been given.

$$g{b}^{\prime }(t)=\min fit(g{b}^j)\hspace{1em}g{b}^j\in \{g{b}^1,g{b}^2,\cdots ,g{b}^{mB}\}$$

(21)

In this, $g{b}^{\prime }(t)$ is the current optimal individual after executing the Brownian motion selection operator; $fit()$ is the fitness function to find each state in the selection space.

5.

Hybrid Algorithm Solution Process

The PSO-CS algorithm is a novel kind of swarm intelligent optimization algorithm that combines the PSO and CS algorithms. It uses the PSO algorithm as its foundation, and its primary parameters and initial values are consistent with those of the classical algorithm. The PSO-CS algorithm’s main steps are stated as follows:

(1)

Set the initial values for the inertia weights and learning factors in the classical particle swarm and cuckoo algorithms; set the initial values for the Lévy flight speed updating operator, the adaptive stochastic offset operator, and the Brownian motion selection operator. These include the Lévy flight speed updating operator’s control parameter $\beta$ and control factor $\mu$, the adaptive stochastic offset operator’s offset control parameter $\kappa$ and the offset threshold ${\lambda }_A$; the Brownian motion step length, population size, and termination conditions; read and store the objective function and constraints;

(2)

Generate an initial population of hybrid algorithms, compute the fitness function value, and store the historical optimal individual $x_{pb}(0)$ and the current global optimal individual $x_{gb}(0)$;

(3)

Calculate the Levy flight coefficients according to Formula (15), which in turn updates the velocities and positions of all individuals based on the Levy flight velocity update operator;

(4)

Judge whether the updated individual satisfies the constraints; if yes, go to step (6); otherwise, go to step (5);

(5)

Adjust the individuals that do not meet the constraints;

(6)

Calculate the fitness function values of all individuals in the hybrid algorithm population and update the historical optimal individual $x_{pb}(t)$ and the current global optimal individual $x_{gb}(t)$;

(7)

Based on the current global optimal individual, combine the Brownian motion step formula to form the potential states of the current optimal individual, construct the state space, calculate the fitness function value corresponding to each state, and update the current global optimal individual optimally;

(8)

Calculate the fitness function value of the hybrid algorithm population to update the historical optimal individual $x_{pb}(t)$ and the current global optimal individual $x_{gb}(t)$;

(9)

Judge whether the termination condition is satisfied; if yes, go to step (12); if not, go to step (3);

(10)

Output the global optimal solution. The PSO-CS algorithm flow is shown in Figure 3.

2.4. PSO-CS Algorithm Tests Based on CEC2022 Test Function Sets

The optimization results of the PSO-CS algorithm are compared with those of the PSO, CS, WOA, SA, COA, and HO algorithms based on the 12 test functions of the CEC2022 optimization function test set to confirm the effectiveness of the PSO-CS method. To ensure fairness in the comparison, each method has a population size of 100, a maximum number of iterations of 1000, and is run separately 20 times. The final optimization result is determined by averaging the 20 optimization outcomes. The accuracy and stability of each method’s optimization are reflected by the optimal value mean and standard deviation, respectively, and the solutions of each algorithm are ranked.

Table 2. Performance comparison between PSO-CS and other algorithms on the CEC2022 test set.

Function	Dim	Values	PSO	PSO-CS	WOA	SA	CS	COA	HO
$F_1$	10	Mean	4.5945 × 103	2.9713 × 102	6.0795 × 103	2.7305 × 104	3.0000 × 102	3.0000 × 102	9.0867 × 103
		Std	1.6852 × 103	0.0000 × 100	5.6259 × 102	1.9669 × 104	0.0000 × 100	0.0000 × 100	2.2343 × 103
		Rank	3	1	4	6	2	2	5
	20	Mean	2.4409 × 104	3.0001 × 102	1.3581 × 103	8.6333 × 104	3.0469 × 102	1.5951 × 105	4.9359 × 104
		Std	5.2925 × 103	2.4612 × 10−2	4.8853 × 102	1.8075 × 104	1.3221 × 100	3.0678 × 10−11	1.3501 × 104
		Rank	4	1	3	6	2	7	5
$F_2$	10	Mean	4.7014 × 10 × 102	4.0004 × 102	4.3611 × 102	5.1237 × 102	4.0607 × 102	5.0172 × 104	7.2524 × 102
		Std	6.0995 × 10	3.2406 × 10−2	3.8456 × 10	1.7181 × 10	2.8569 × 100	2.4964 × 102	1.9308 × 102
		Rank	4	1	3	5	2	7	6
	20	Mean	7.9187 × 102	4.0000 × 102	4.5007 × 102	9.6509 × 102	4.1323 × 102	7.2730 × 103	1.6328 × 103
		Std	7.7825 × 10	2.5139 × 10	3.3160 × 10	2.4981 × 102	1.5138 × 10	0.0000 × 100	3.9209 × 102
		Rank	4	1	3	5	2	7	6
$F_3$	10	Mean	6.2905 × 102	6.0313 × 102	6.2463 × 102	6.7429 × 102	6.0417 × 102	6.9074 × 102	6.3250 × 102
		Std	1.7356 × 100	8.7382 × 10−1	1.7667 × 10	2.6685 × 100	0.0000 × 100	0.0000 × 100	2.0709 × 100
		Rank	4	1	3	6	2	7	5
	20	Mean	6.4835 × 102	6.0321 × 102	6.5840 × 102	7.1149 × 102	6.1815 × 102	7.4688 × 102	6.7073 × 102
		Std	5.3247 × 100	4.8070 × 100	1.4006 × 10	9.1742 × 100	3.8591 × 100	1.1984 × 10−13	7.8182 × 100
		Rank	3	1	4	6	2	7	5
$F_4$	10	Mean	8.4823 × 102	8.1559 × 10	8.4278 × 102	9.1932 × 102	8.1940 × 102	9.0608 × 102	8.3458 × 102
		Std	7.9324 × 100	2.7722 × 100	1.9700 × 10	8.8567 × 100	2.1106 × 100	0.0000 × 100	1.4224 × 101
		Rank	5	1	4	7	2	6	3
	20	Mean	9.5769 × 102	8.6786 × 102	9.1931 × 102	1.0924 × 103	8.9122 × 102	1.1106 × 103	9.5849 × 102
		Std	9.8838 × 100	2.2900 × 101	3.4061 × 101	2.1452 × 101	7.6080 × 100	2.3967 × 10−13	1.7615 × 101
		Rank	4	1	3	6	2	7	5
$F_5$	10	Mean	1.1892 × 103	9.0531 × 102	1.3677 × 103	1.9351 × 103	9.0638 × 102	3.8096 × 103	1.1502 × 103
		Std	5.8726 × 101	0.0000 × 100	3.7566 × 101	5.3324 × 102	2.8555 × 100	0.0000 × 100	2.2494 × 102
		Rank	4	1	5	6	2	7	3
	20	Mean	4.3709 × 103	1.4389 × 103	3.5909 × 103	8.3292 × 103	1.4948 × 103	5.5721 × 103	3.0879 × 103
		Std	5.8027 × 102	3.3996 × 102	1.2000 × 103	1.4561 × 103	6.9299 × 101	9.5869 × 10−13	5.1781 × 102
		Rank	5	1	4	7	2	6	3
$F_6$	10	Mean	9.0851 × 105	1.8008 × 103	5.0653 × 103	6.6556 × 103	6.9061 × 103	2.1947 × 108	2.2076 × 106
		Std	4.0209 × 105	3.3110 × 10−1	9.7850 × 102	1.9873 × 103	7.3874 × 101	0.0000 × 100	3.1119 × 106
		Rank	5	1	2	3	4	7	6
	20	Mean	1.1266 × 108	1.8440 × 103	8.1254 × 103	1.1053 × 105	1.0345 × 104	1.0197 × 1010	6.2254 × 108
		Std	8.1138 × 10 7	4.0885 × 100	7.9809 × 103	2.7820 × 105	9.5156 × 103	2.0105 × 106	2.9250 × 108
		Rank	6	1	2	4	3	5	7
$F_7$	10	Mean	2.0510 × 103	2.0272 × 103	2.0684 × 103	2.1363 × 103	2.0298 × 103	2.2848 × 103	2.0770 × 103
		Std	4.2911 × 100	1.3434 × 100	1.2496 × 101	1.3888 × 101	1.4291 × 100	2.0497 × 101	4.0105 × 101
		Rank	3	1	4	5	2	6	7
	20	Mean	2.1344 × 103	2.0698 × 103	2.1733 × 103	2.3343 × 103	2.0885 × 103	2.5103 × 103	2.1858 × 103
		Std	1.2132 × 101	2.9131 × 101	4.3172 × 101	4.2579 × 101	5.4936 × 100	4.7935 × 101	4.3149 × 101
		Rank	3	1	4	6	2	7	5
$F_8$	10	Mean	2.2295 × 103	2.2114 × 103	2.2293 × 103	2.2459 × 103	2.2209 × 103	2.9437 × 103	2.2351 × 103
		Std	1.8604 × 100	1.0935 × 100	5.1993 × 100	6.3473 × 101	1.2369 × 100	1.3207 × 102	8.0561 × 100
		Rank	4	1	3	6	2	7	5
	20	Mean	2.3684 × 103	2.2339 × 103	2.3328 × 103	2.3181 × 103	2.2519 × 103	3.3954 × 104	2.4012 × 103
		Std	1.5466 × 101	1.1876 × 100	1.2793 × 102	1.9100 × 101	7.8982 × 10−1	2.5153 × 104	1.7064 × 102
		Rank	6	1	5	4	3	7	2
$F_9$	10	Mean	2.6231 × 103	2.5273 × 103	2.5294 × 103	2.7251 × 103	2.5289 × 103	4.0536 × 103	2.6748 × 103
		Std	1.1439 × 102	1.1962 × 10−1	1.3342 × 10−2	1.0071 × 102	7.4448 × 10−2	2.2546 × 102	4.7994 × 101
		Rank	4	1	2	6	3	7	5
	20	Mean	2.5549 × 103	2.4808 × 103	2.4829 × 103	3.0348 × 103	2.4809 × 103	5.1532 × 103	2.9109 × 103
		Std	4.6963 × 101	1.0868 × 10−3	2.2575 × 100	1.6038 × 102	2.3199 × 10−2	6.9633 × 102	5.9396 × 101
		Rank	4	1	3	6	2	7	5
$F_{10}$	10	Mean	2.5078 × 103	2.5004 × 103	2.5688 × 103	2.6107 × 103	2.5009 × 103	4.9524 × 103	2.5165 × 103
		Std	1.1964 × 100	1.0658 × 10−2	9.6564 × 10	1.0047 × 102	7.2031 × 10−2	3.9130 × 10	1.1755 × 101
		Rank	3	1	5	6	2	7	4
	20	Mean	5.5576 × 103	2.5008 × 103	5.2629 × 103	6.2856 × 103	2.5009 × 103	8.7814 × 103	5.1262 × 103
		Std	2.5691 × 102	1.0372 × 10−1	5.1585 × 102	1.1019 × 103	3.7636 × 10	5.9494 × 10	1.7837 × 103
		Rank	6	1	3	4	2	7	5
$F_{11}$	10	Mean	2.7946 × 103	2.6000 × 103	2.8753 × 103	3.1918 × 103	2.7504 × 103	5.1052 × 103	2.8180 × 103
		Std	1.4312 × 10	2.8584 × 10−9	1.7655 × 102	4.9260 × 102	3.5493 × 10−9	0.0000 × 100	4.2883 × 10
		Rank	3	1	5	6	2	7	4
	20	Mean	5.1297 × 103	2.9251 × 103	3.0096 × 103	9.5241 × 103	2.9500 × 103	1.0783 × 104	6.6099 × 103
		Std	1.6095 × 103	1.2504 × 102	1.0273 × 102	1.1613 × 103	7.0711 × 10	0.0000 × 100	4.1787 × 102
		Rank	4	1	3	6	2	7	5
$F_{12}$	10	Mean	2.8651 × 103	2.8589 × 103	2.8696 × 103	2.9014 × 103	2.8626 × 103	2.9907 × 103	2.9887 × 103
		Std	1.6179 × 10−1	1.6050 × 10−1	8.0333 × 100	4.9578 × 100	5.2891 × 10−1	4.3252 × 10	4.1039 × 10
		Rank	3	1	4	5	2	7	6
	20	Mean	2.9702 × 103	2.9474 × 103	3.0309 × 103	3.2915 × 103	2.9510 × 103	3.5523 × 103	3.3798 × 103
		Std	2.0820 × 10	7.9699 × 10−1	1.5233 × 10	9.7613 × 10	7.6760 × 100	2.3911 × 10	2.5929 × 102
		Rank	3	1	4	6	2	7	5

displays the outcomes of the comparison.

Table 2 shows that, when it comes to the mean and standard deviation of the optimal values that are acquired after 1000 iterations of solving, the PSO-CS method performs better than the other six algorithms. The PSO-CS algorithm works particularly well on the functions $F_8$, $F_{10}$ and $F_{12}$. Furthermore, the PSO-CS algorithm receives the highest score among all the functions that are examined, indicating that it is more successful at enhancing the PSO algorithm’s optimization.

The convergence curves of PSO-CS compared to other algorithms on the CEC2022 test set are presented as indicated in Figure 4 to visualize the convergence performance of the PSO-CS algorithm. When compared to other methods, PSO-CS exhibits faster and more accurate convergence in the majority of the test functions. This suggests that the optimization technique used in this research can successfully increase PSO’s convergence speed and accuracy, leading to superior optimization outcomes. At the same time, other algorithms easily fall into local optimization, which leads to slower convergence or even stagnation.

In addition to analyzing the numerical results obtained by the PSO-CS algorithm and other algorithms, the superiority of the PSO-CS algorithm is demonstrated by two non-parametric statistical tests, the Friedman test and the Bonferroni–Dunn test.

A multiple comparison test called the Friedman test is used to see whether there are any appreciable variations between algorithms. The Friedman test’s algorithm ranking principle states that the algorithm with the highest performance is ranked the smallest, while the algorithm with the lowest performance is ranked the largest. The optimal values and standard deviations produced by various methods in Table 2 were subjected to the Friedman test; the computed outcomes are displayed in

Table 3. Friedman’s test results for PSO-CS and other algorithms based on solving for the mean and standard deviation of the optimal values in Table 2 (best rankings are in bold).

		Mean of Optimal Values	SD
	N	12	12
Test results	Chi-square	57.358	59.964
	p-value	<0.01	<0.01
	PSO	3.83	4.33
	PSO-CS	1.00	1.00
	WOA	3.58	3.42
Friedman test value	SA	5.71	5.50
	CS	2.29	2.17
	COA	6.42	6.75
	HO	5.17	4.83

.

Friedman’s test results in Table 3 show that all p-values are below the significance level of α = 0.01, suggesting that these seven algorithms differ significantly from one another. The PSO-CS algorithm performs the best in terms of optimizing the solution, followed by the CS, WOA, and other algorithms. The test results of the mean and standard deviation of the optimal values also show this. Therefore, it can be said that the PSO-CS method has greater accuracy than the other algorithms. Nevertheless, the Bonferroni–Dunn test is used to assess the optimization performance of the PS-FW algorithm because Friedman’s test can only determine whether there is a significant difference between all algorithms from an overall perspective and cannot specifically compare the performance difference between the PSO-CS algorithm and each other algorithm.

Finding significant differences between two or more algorithms may be achieved quite simply with the help of the Bonferroni–Dunn test. The following equation determines the critical difference, which must be exceeded by the performance rankings of the two algorithms for there to be a significant difference between them for the Bonferroni–Dunn test [26].

$$C{D}_{\alpha }=q_{\alpha }\sqrt{{\displaystyle \frac{N_i(N_i+1)}{6N_f}}}$$

(22)

In this formula, $N_i$ and $N_f$ are the number of algorithms and benchmark functions, respectively, $q_{\alpha }$ is the critical value at the significance level $\alpha$, which is as follows at different significance levels:

$$q_{0.05}=2.77,q_{0.1}=2.54$$

(23)

Combining Formulas (22) and (23) yields the following critical differences at different significance levels:

$$C{D}_{0.05}=2.44,C{D}_{0.1}=2.24$$

(24)

The Bonferroni–Dunn test was conducted on the optimal value mean and optimal value standard deviation, utilizing the ranks acquired from Friedman’s test. The histograms based on the optimal value mean and optimal value standard deviation are plotted as shown in Figure 5. For more intuitively displaying the results of the Bonferroni-Dunn test combined with the critical difference. The pink horizontal line represents the ranking of the best algorithms, and the two black and red horizontal lines, respectively, correspond to the threshold level values under the significance level of $\alpha =0.05$ and $\alpha =0.1$, which are equal to the sum of the values for the critical difference and the lowest ranked value. An algorithm is shown to perform worse than the optimal ranking algorithm if the height of its histogram is greater than the threshold level’s horizontal line. The PSO-CS algorithm’s histogram is the lowest of all the algorithms, as shown in Figure 5a. The PSO-CS algorithm outperforms the PSO, WOA, SA, COA, and HO algorithms at both the 90% and 95% significance levels. In addition, PSO outperforms CS but non-significantly. This indicates that the PSO-CS algorithm incorporates the correctness of the three optimization operators and performs well in terms of global optimization solving power. Figure 5b shows that the PSO-CS algorithm has relatively better robustness compared to other improved swarm intelligence optimization algorithms. It outperforms PSO, WOA, SA, COA, and HO algorithms for the PSO-CS algorithm at 90% and 95% confidence levels. It shows that the PSO-CS algorithm is significantly better than PSO, WOA, SA, COA, and HO algorithms in terms of the stability performance of the algorithm in solving the optimization problem. CS algorithm is also better in terms of robustness, so the PSO-CS algorithm does not significantly outperform the NRBO algorithm in the Bonferroni–Dunn test.

3. Results and Discussions

3.1. Scheduling Optimization Results for Sewage System on a Certain Day

3.1.1. Scheduling Optimization Scheme

The sewage and injection stations for this sewage system, as well as the water pipeline paths and pipeline distances between them, are shown in

Table 4. Sewage system scheduling optimization scheme.

Name of Sewage Station	Name of Water Injection Station	Pipeline Path	Length (m)
NIII;-1D	ZN3	NIII;-1D→ZN3	2600
NIII;-1D	ZN11	NIII;-1D→ZN11	3100
NIII;-1D	ZN16-P	NIII;-1D→ZN16-P	1100
NIII;-1D	ZN17-P	NIII;-1D→ZN17-P	1200
NIII;-1	ZN8	NIII;-1→ZN8	1207
NIII;-1	ZN7	NIII;-1→ZN7	2250
NIII;-1	ZN19	NIII;-1→ZN19	500
NIII;-1	ZN10	NIII;-1→ZN10	1382
NII-2-G	ZNII-4	NII-2-G→ZNII-4	5150
NII-2-G	ZN22	NII-2-G→ZN22	1350
NII-2	ZNII-4	NII-2→ZNII-4	5240
NII-2	ZN8	NII-2→ZN8	650
NII-2	ZN22	NII-2→ZN22	340
NII-2	ZN11	NII-2→ZN11	2390
N16	ZNII-4	N16→ZNII-4	2550
N16	ZN22	N16→ZN22	920
N16	ZN7	N16→ZN7	800
N16	ZN11	N16→ZN11	1100
N16	ZN16-P	N16→ZN16-P	70
N11-D	ZN11	N11-D→ZN11	50
N11-D	ZN16-P	N11-D→ZN16-P	820
NII-7 T	ZNII-4	NII-7 T→ZNII-4	2750
NII-7 T	ZN22	NII-7 T→ZN22	1925
NII-7 T	ZN16-P	NII-7 T→ZN16-P	1100
NII-7 D	ZN16-P	NII-7 D→ZN16-P	1000
NII-7 D	ZN17-P	NII-7 D→ZN17-P	980
NIII;-2D	ZNIII;-2	NIII;-2D→ZNIII;-2	60
NIII;-2D	ZN16-P	NIII;-2D→ZN16-P	3500
SN2801	ZN23	SN2801→ZN23	2780
SN2801	ZN9	SN2801→ZN9	600
N20-D	ZN20	N20-D→ZN20	1378
N4	ZN20	N4→ZN20	850
N4	ZN7	N4→ZN7	230
N17	ZNIII;-2	N17→ZNIII;-2	1750
N17	ZN8	N17→ZN8	800
N17	ZN7	N17→ZN7	450
N17	ZN10	N17→ZN10	1200
N17	ZN17-P	N17→ZN17-P	1350
NIII;-7T	ZN8	NIII;-7T→ZN8	1350
NIII;-7T	ZN7	NIII;-7T→ZN7	2110
NIII;-7T	ZN17-P	NIII;-7T→ZN17-P	800
N5-P	ZN5	N5-P→ZN5	780
N5	ZN7	N5→ZN7	2556
N5	ZN10	N5→ZN10	4200
N5	ZN5	N5→ZN5	1378
NII-1	ZNII-1	NII-1→ZNII-1	190
NII-1	ZNII-4	NII-1→ZNII-4	3125
NII-1	ZN23	NII-1→ZN23	436
SN25-D	ZNIII;-2	SN25-D→ZNIII;-2	450
SN25-D	ZN20	SN25-D→ZN20	5000
SN25-D	ZN17-P	SN25-D→ZN17-P	500
N12-D	ZN12	N12-D→ZN12	140
N12-D	ZN13	N12-D→ZN13	3360
N12-D	ZN12-P	N12-D→ZN12-P	1340
N15	ZNII-1	N15→ZNII-1	1632
N15	ZN8	N15→ZN8	896
N15	ZN9	N15→ZN9	1750
N15	ZN3	N15→ZN3	3840
N15	ZN14	N15→ZN14	2000
N15	ZN12-P	N15→ZN12-P	530
NIII;-6T	ZN8	NIII;-6T→ZN8	1280
NIII;-6T	ZN3	NIII;-6T→ZN3	1060
N6-P	ZN6-P	N6-P→ZN6-P	580
N6-P	ZN6-W	N6-P→ZN6-W	780
N6-P	ZN19	N6-P→ZN19	3136
N3	ZN6-W	N3→ZN6-W	3206
N3	ZN9	N3→ZN9	2750
N3	ZN3	N3→ZN3	300
N3	ZN10	N3→ZN10	2800
N6-D	ZN6-W	N6-D→ZN6-W	2340
N6-D	ZN19	N6-D→ZN19	575
N6	ZN6-P	N6→ZN6-P	1720
N6	ZN6-W	N6→ZN6-W	630
N13-P	ZNII-1	N13-P→ZNII-1	6890
N13-P	ZNII-4	N13-P→ZNII-4	6920
N13-P	ZN23	N13-P→ZN23	579
N13-P	ZN13	N13-P→ZN13	950
N13-P	ZN14	N13-P→ZN14	920
N13-ND	ZN12	N13-ND→ZN12	3565
N13-ND	ZN13	N13-ND→ZN13	1300
N13-D	ZN23	N13-D→ZN23	2205
N13-D	ZN13	N13-D→ZN13	380

.

The sewage system operating model is solved using the hybrid PSO-CS optimization technique presented in this research, taking into account the production data of the sewage system on that particular day. The sewage system’s pre- and post-optimization water scheduling schemes are depicted in Figure 6 and Figure 7, respectively. It is clear from these that the optimized water transfer scheme has undergone significant modification in comparison to the pre-optimization one. The improved water transfer strategy is more focused on the injection station’s water supply, whereas the pre-optimization water transfer scheme has scattered distribution and a lower water volume. Using the sewage station “NIII-1” as an example, water is sent to injection stations downstream in four directions: “ZN8”, “ZN7”, “ZN19”, and “ZN10”. The water supply plan before optimization indicates that “NIII-1” exclusively provided water to “ZN8”, “ZN7”, and “ZN19”. Following optimization, “NIII-1” solely provides water to the “ZN10” injection station; the volume of water supplied is 6559 m³, greater than that of “ZN19”. The water amount is 6559 m³, which is more than it was before optimization, and “NIII-1” solely provides water to the “ZN10” injection station after optimization. This demonstrates how the optimized scheduling technique successfully raises operating efficiency, lowers energy consumption, and enhances the sewage system’s water scheduling strategy.

3.1.2. Comparison and Result Analysis before and after Optimization

Based on the water scheduling results for the sewage system, compute the energy consumption of the sewage system scheduling before and after optimization, as illustrated in Figure 8. The total operational energy consumption of the optimized system is 712.84 × 10⁶ m⁵/d, and the operational energy consumption of the optimized sewage system is reduced by 19% when compared to the system operation before optimization. The total operational energy consumption generated by the scheduling plan on that day is 879.95 × 10⁶ m⁵/d.

Table 5. Water requirements for each injection station for the day.

Number	Name	Water Demand (m3/d)
1	ZNII-1	8352
2	ZNII-4	0
3	ZNIII;-2	9997
4	ZN8	14,795
5	ZN22	9795
6	ZN23	9548
7	ZN20	16,415
8	ZN9	7109
9	ZN6-P	0
10	ZN6-W	7908
11	ZN7	14,563
12	ZN3	9008
13	ZN12	7068
14	ZN19	5930
15	ZN13	16,825
16	ZN14	7628
17	ZN11	9259
18	ZN10	7321
19	ZN5	7558
20	ZN12-P	11,456
21	ZN16-P	5276
22	ZN17-P	4383

displays the volume of water needed for the production and operation of each injection station on that particular day. The water supply for each injection station can be provided by the scheduling plan before and during optimization, as seen in Figure 9. For example, “ZN19” and “ZNII-1” require 5930 m³ and 8352 m³, respectively, and the water supply from the upstream sewage station before the optimization is 3802 m³ and 4640 m³, respectively. These amounts represent 64.1% and 55.6% of the required water quantity. Based on the optimized scheduling plan, the water supply from the injection stations “ZN19” and “ZNII-1” is 5930 m³ and 8352 m³, respectively, which accounts for 100% of the injection stations’ water demand. This indicates that the optimized scheduling plan can effectively meet the injection stations’ water demand and can support the water injection system’s efficient operation.

There were 27 sewage stations produced and running on that day. The water quantity entering and exiting the sewage stations was determined based on the day’s production statistics and optimization outcomes, as indicated in

Table 6. Volume of water entering and leaving the sewage station.

Number	Name	The Volume of Incoming Water for the Day (m3)	Water Output for the Day (m3)	Optimized Water Output (m3)
1	NII-1	10,833	4821	8352
2	NII-2	11,847	5545.55	9795
3	NII-7D	7110	2249	5582
4	NII-7T	9673	0	0
5	NIII-1D	15,287	0	0
6	NIII-1	6559	5488	6559
7	NIII-2D	4201	3807	3201
8	NIII-6T	6167	0	0
9	NIII-7T	16,581	3552.05	16,581
10	N20-D	3761	3226	2021
11	N6-P	9844	0	0
12	N6	10,372	0	7908
13	N6-D	7784	1120	5930
14	N3	10,927	9433	9008
15	N12-D	9566	3856	7068
16	N17	22,768	12,652.4	20,911
17	N13-D	6988	2791	6988
18	N13-P	18,462	8635	18,462
19	N13-ND	7611	5278	5841
20	N15	14,688	14,080	13,688
21	N11-D	8563	3881.3	6563
22	N4	14,394	7532	14,394
23	N5-P	7281	2000	7281
24	N5	9647	2000	277
25	N16	9755	8242	2755
26	SN25-D	2442	1746	1442
27	SN2801	7587	3213	7587

. The water quantity in and out of the sewage stations was not adequately balanced by the current scheduling plan, which prevented some of the inlet water from being effectively transferred to the downstream injection station. This led to the hoarding of water in the sewage stations. Before optimization, the water quantity out of the sewage stations was typically lower than that of the inlet water. The amount of water in and out of 7 sewage stations reaches a balanced state after optimization, and the amount of water out of the sewage station rises significantly, with the majority of the water out of the sewage station being close to or equal to the amount of inlet water. This indicates that the water transfer scheme effectively improves the utilization rate of water resources in the sewage stations, and the oilfield sewage system’s operation efficiency has been greatly enhanced.

3.2. Optimization Results of Sewage System Scheduling for a Particular Month

The production data of the plant in June is utilized as the dataset to optimize the water transfer scheme for 30 production days of the month to confirm the stability of the PSO-CS optimization algorithm. Figure 10 displays the injection stations’ water supply both before and after optimization. There are few injection stations that meet the water requirement before optimization. In particular, on Days 9, 10, and 18, the water transfer scheme only provides for the water requirements of 15 injection stations, or 68.18% of the total number of injection stations. This has an impact on the downstream injection system’s ability to operate efficiently. Furthermore, the optimized water transfer scheme demonstrates that the number of water injection stations meeting the water requirement is 22 per day, which can effectively ensure the normal operation of the downstream water injection system. The above analysis demonstrates that the water scheduling scheme using human experience cannot adapt to the changes in the system operating conditions, resulting in the inability to effectively meet the water requirements of the injection stations. Using the intervals from Day 17 to Day 18 and Day 10 to Day 11 as examples, the water transfer scheme also demonstrates that the number of injection stations meeting the water quantity requirement fluctuates greatly during these two periods, indicating that the stability of the water transfer scheme is low. The ideal water supply scheme can be modified in response to modifications in system operating conditions, guaranteeing the stability of the scheduling decision. This is because the scheduling scheme produced by PSO-CS optimization does not depend on subjective considerations.

The total energy used by the sewage system for each day of operation, both before and after the optimization in that particular month, is displayed in Figure 11. The sewage system’s energy consumption ranges from 800 × 10⁶ m⁵/d to 950 × 10⁶ m⁵/d before optimization. Following optimization, the energy consumption remains within the same range, ranging from 650 × 10⁶ m⁵/d to 750 × 10⁶ m⁵/d, indicating a notable reduction in comparison to the pre-optimization levels. This demonstrates that the sewage system that has been adjusted may effectively reduce its energy usage while maintaining stable operation.

4. Conclusions

This paper establishes a sewage system scheduling optimization model by taking the actual production and operation of the sewage system of an oil extraction plant in the Daqing oilfield as an example. Aiming at the nonlinear, multi-constraint, and non-convex characteristics of the oilfield wastewater system scheduling model. The improved particle swarm optimization algorithm PSO-CS, which integrates the cuckoo optimization algorithm and particle swarm optimization algorithm, is proposed to solve this complex problem, and the optimal water quantity scheduling strategy of the sewage system is obtained. The main findings of this study are as follows:

(1)

The optimized sewerage system water supply direction and quantity were effectively adjusted. This indicates that the optimized water scheduling strategy can effectively improve the operation of the system;

(2)

The operational energy consumption of the sewage system was effectively reduced after optimization. The total operational energy consumption generated by the scheduling program on the same day was reduced from 879.95 × 10⁶ m⁵/d to 712.84 × 10⁶ m⁵/d, and the operational energy consumption of the sewage system after optimization was 19% lower than that of the system before optimization;

(3)

Significant increase in water output from sewage stations after optimization, with the number of sewage stations in and out of the water balance rising to 7 compared to 0 before optimization and the rate of meeting the water requirements of downstream water injection stations reaching 100%;

(4)

Monthly scheduling optimization of the sewage system has a significant reduction in energy consumption. Before optimization, the energy consumption of the sewage system in June was 800 × 10⁶ m⁵/d–950 × 10⁶ m⁵/d. After optimization, the energy consumption was 650 × 10⁶ m⁵/d–750 × 10⁶ m⁵/d, and the energy consumption of the sewage system operation after optimization was significantly reduced.

Although this research significantly improves the optimization of water scheduling in oilfield sewage systems by improving the PSO algorithm, there are still some shortcomings: the computational complexity of the improved algorithm may be higher when dealing with very large-scale systems, and it needs to be further optimized to improve the computational efficiency. Future research can further explore and develop more efficient optimization algorithms to cope with larger-scale and more complex scheduling problems.