Optimization of pump scheduling in waterworks considering load balancing using improved genetic algorithm

Abstract

The water transmission and distribution process of the water supply pump station of the water purification plant plays a key role in the entire urban water supply system. When the requirements of water supply quantity and water pressure are satisfied, the reduction of operating energy consumption of the pump set and improvement of its service life are urgent problems. Therefore, to reduce the cost of water supply pump station, a mathematical model is established to minimize the energy consumption of pump group, the number of pump switches and the load balancing in this paper. In order to solve the pump scheduling problem, a two-stage strategy based on genetic algorithm is proposed. In stage one, the frequency conversion ratio and the number of pumps needed to be turned on at the lowest energy consumption are calculated. In stage two, through the improved genetic algorithm and iterative way to reduce the number of pump switches and load balancing. Finally, a case study from a real waterworks in Suzhou, China is used to verify the validity of the proposed model. Numerical results reveal that the improved genetic algorithm outperforms the competing algorithms. In addition, a proper sensitivity analysis allows assessing the effects under different pump operating conditions.

Keywords

Pump scheduling load balancing improved genetic algorithm

1 Introduction

As shown in Fig. 1, the treatment process of the water plant is to pump the raw water through the pump into the pool, and then execute a series of physical and chemical reactions on it [1]. Develop a water production plan by forecasting water demand [2 –4], the water that meets the production standards will be stored in the savings pool waiting for distribution [5], and then the pump group in the water supply pump station pump the water to the municipal pipe network. Although different water purification plants have different scales and different water production processes, the highest proportion of electricity consumption is in the water supply pump station. So, it is necessary to optimize the pump system to save the cost [6 –8]. Hence, the aim of this study is to optimize the scheduling scheme of water supply pump station to reduce the power consumption, increase the service life of pumps and balance the differences between pumps.

Fig. 1

Water production process of water purification plant.

In the past decades, pump scheduling has received more and more attention from academia and enterprises. The methods used to solve the pump scheduling problem in previous literature mainly include exact algorithm and heuristic algorithm [9 –13]. Cantu-Funes et al. proposed a nonlinear, non-convex expression and high-performance heuristic to solve the pump scheduling optimization problem, and they obtained the lowest cost pump scheduling scheme under the premise of satisfying hydraulic constraints [14]. Puleo et al. proposed a method based on linear programming to quickly obtain an optimal pump scheduling scheme. In their model, taking continuous pump flow rate as the decision variable, it was converted into discrete pump flow rate with a period of 24 hours [15]. Although these methods are relatively accurate and interpretable, the requirement for the explicit formulations of the scheduling process makes them difficult to apply in complex water supply systems. Therefore, heuristic algorithm begins to gradually get the attention of researchers due to their better prediction accuracy and without need the specific expert knowledge. Hooshmand et al. described the pump scheduling problem as a mixed integer nonlinear programming model, and a two-phase method was presented to solve the problem. An initial feasible solution was generated by a heuristic method based on the underlying structure in the first stage, then it was input in the second stage to achieve an approximate optimal solution [16]. Ormsbee et al. developed a heuristic method to solve the operation problem of the lowest cost pump in the water distribution system considering the nonlinear characteristics of water distribution network [17]. Zhuan et al. established a dynamic programming model for the pump station with the aim of minimizing the operation cost of the pump station, and two dynamic programming models of the cascade pump station were established according to the flow delay time between stages [18]. Although these methods can effectively reduce the pump running power consumption, they can be time-consuming to deal with large scale pump scheduling problems. To solve this problem, meta-heuristic algorithm has been used to solve the pump scheduling problem. Jafari-Asl et al. proposed a binary dragonfly algorithm with a new transfer function for pump scheduling with the goal of reducing total energy consumption costs [19]. Ngancha et al. modeled and simulated large volume pumping and purification processes from the perspective of pumping energy consumption, and they proposed an optimal energy consumption rate management system to achieve minimum energy consumption at any given load [20]. Luca et al. adapted the model based on a variety of natural multi-population optimization algorithms, and the main objective was to obtain an optimal operational schedule for each set of pumps with the least amount of wasted energy [21]. Giacomello et al. proposed a hybrid optimization method to solve the pump scheduling problem using linear programming and greedy algorithms [22]. The above studies only focus on the single objective of pump energy consumption, but the problem of pump scheduling optimization includes not only power consumption but also other optimization objectives. Yasaman et al. established a multi-objective mathematical model with energy cost and pump switch number, and used the adaptive NSGA-II algorithm to solve the pump scheduling problem [23]. Lopez-Ibanez et al. developed an ant colony optimization model to solve the problem of optimal pump control considering the pump switch number [24]. Francesco et al. proposed a multi-objective Harmony-Search algorithm for the pump scheduling problem which took the energy cost and the number of pump switches as targets [25]. Zhuan et al. proposed an extended reduced dynamic programming algorithm for optimal operation scheduling problem of multi-pump pumping stations, both the energy cost and the maintenance cost were considered in the performance function of the optimization problem [26]. These scholars mainly focused on the importance of the number of pump switches on pump life and did not consider the balance of pump service time, while load balancing can effectively improve the reliability and safety of the pump group. Pump running time is too long will more likely to lead to failure, so it is necessary to balance the working time of each pump.

To sum up, the operation cost optimization of water supply pump station is mainly to reduce the pump’s electric energy consumption and operation and maintenance costs. Therefore, a scheduling model is established with three functions of the minimum total power consumption, the minimum number of pump switches, and minimize pump group load balancing in this paper. The effects of time-of-use electricity price, water depth, continuous pump operation time and the difference of water demand index in each area are considered in the mathematical model and constraint conditions. For the above pump scheduling optimization objectives, a two-stage strategy is presented to solve the pump scheduling problem. In the first stage, the results of frequency conversion ratio and the number of pumps need to turned on under the lowest energy consumption are obtained by using genetic algorithm. In the second stage, improved genetic algorithm and iterative way are used to optimize the number of pump switches and load balancing. Finally, the model in this paper is applied to the real pipe network, and the results fully prove the feasibility of the model and the solving method. In addition, even if some pumps are faulty, the pump scheduling can still run reliably, which reflects the robustness of this model and solution method.

The rest of the sections of this article are as follows. In Section 2, the pump scheduling problem is described and some formulas are derived. In Section 3, mathematical modeling and related constraints are presented. Section 4 describes the two-stage strategy for solving pump scheduling based on genetic algorithm. In Section 5, the model and method are applied to the real case to get the pump scheduling scheme. Section 6 presents and analyzes the experimental data. Finally, Section 7 makes a summary of this paper.

2 Problem statement and mathematical formula

2.1 Problem statement

As shown in Fig. 2, the water supply pump station mainly includes savings pool, water supply pump station and several pipe networks. Water supply pump station is composed of multiple pumps in parallel. The difference of frequency conversion ratio and number of pumps in parallel state will directly affect the consumption of electric energy. Therefore, it is necessary to study the performance of multiple pumps in parallel.

Fig. 2

The configuration of water supply pump station.

The performance parameters of all pumps in the water supply pump station are the same, and the pumps are divided into two categories: common pumps and flexible pumps. Common pump can only supply water to one network, while flexible pump can supply water to different networks by switching the intelligent switch. In the 24-hour cycle time of the day, the pump scheduling problem is defined as rearranging the open or closed states of all pumps at different times after obtaining the frequency conversion ratio and the number of pumps on at the lowest power consumption, and then each pump uses a binary string of 0 or 1 to indicate whether it was off or on within 24 hours. To simplify the study, assume that the water demand and pressure at each time for all areas within a 24-hour period are known, and adjust the pump scheduling state every hour. For the convenience of understanding the formula in this paper, the relevant symbols and descriptions are shown in Table 1.

Table 1

Notation

Description
Parameters
J	Region number
M	Total number of pumps
N	The number of flexible pumps
T	Set of time periods
Q _N	Water flow of variable-speed pump under rated power
h	Pump height compared relative to ground
k	The amount of energy needed to lift one meter per cubic meter of water
γ	Water weight
σ	Threshold of water pressure
β_min, β_max	Threshold value of efficiency
T _max	The maximum continuous operating time of the pump
ρ	water density
g	Gravitational acceleration
a₁, a₂, a₃	Flow-efficiency fitting parameters
b₀, b₁, b₂	Flow-head fitting parameters
c₁, c₂, c₃	Power-speed ratio fitting parameters
Variables
S _ti	The frequency conversion of pump i during time t
S_i,min, S_i,max	The frequency conversion of pump i
Q _ti	The water flow of pump i during time t
Q_i,min, Q_i,max	Water flow threshold of pump i
Q _pret,j	Total water demand in Area j during time t
Q _tj	Total water flow in Area j during time t
Q _pool,t	The total amount of water in the savings pool
H _p	Pump head
H_i,min, H_i,max	Head threshold of pump i
H _jt	Water head of delivery in Area j during time t
H _pre,jt	The predicted value of water head of delivery in Area j during time t
T _i	Continuous working hours of pump i
R	All pumps’ number of switches
R _(t,i)	The state variable of pump i during time t, 0 or 1
P _ti	The power of the pump i at t moment
Q _t-all	Total water flow during time t
X _i	Total working time of pump i during period
$\bar{X}$	Mean value of each pump working time
L	The value of load balance
t	specific moment
C	Total power consumption cost
e _t	Price of electricity at time t
Decision variable
w _ti	On or off state, 1 or 0
w _c,j	Open and closed state of pump c in Area j, 0 or 1
h _t	Water surface of fresh pool relative to ground during time t
n _i	Pump rotate speed of pump i
n_i,min, n_i,max	Pump rotate speed threshold of pump i
η	Efficiency (0-1)

2.2 Mathematical formula

As shown in Fig. 3, the use efficiency and pump head of a single variable-speed pump will change with the difference in the controlled effluent flow. When the variable-speed pump is working at the rated speed, the head (water pressure) of it will decrease with the increase of the set flow rate, and the efficiency will generally increase with the increase of the flow rate. The expressions of the fitting curves of η_N and H_N are shown in Equation (1) and Equation (2).

Fig. 3

Single pump flow-efficiency/head fitting curve.

$η_{N} = a_{1} Q_{N} + a_{2} Q_{N}^{2} + a_{3} Q_{N}^{3}$ (1) $H_{N} = b_{0} + b_{1} Q_{N} + b_{2} Q_{N}^{2}$ (2)

Where, a₁, a₂, a₃ is the fitting coefficient of the efficiency; b₀, b₁, b₂ is the fitting coefficient of the head; η_N is efficiency at rated power; H_N is the head at rated power; Q_N is the independent variable of water flow at rated power. As shown in Fig. 3, the water supply index of the maximum efficiency point in the optimal state can be obtained as point (Q_A, H_A). The fitting parameters in Equation (1) and Equation (2) can be obtained through least squares fit [27, 28].

According to laws of similitude, the relationship between S and Q (and H) is shown in Equation (3): $S = \frac{Q}{Q_{N}} = \frac{n}{n_{N}}, \frac{H}{H_{N}} = S^{2}$ (3)

Q_N, n_N, H_N are the relevant values at the rated speed, it can be deduced that: $η = \frac{a_{1} Q}{S} + \frac{a_{2} Q^{2}}{S^{2}} + \frac{a_{3} Q^{3}}{S^{3}}$ (4) $H = b_{0} S^{2} + b_{1} SQ + b_{2} Q^{2}$ (5)

Equation (6) represents the relationship between efficiency and total shaft power of the pump, when the efficiency of the pump reaches the highest, the electric power consumed by the water pump is the smallest, Q_p, H_p is the water flow and head of the water supply index. $η = \frac{ρ {gQ}_{p} H_{p}}{N_{p}} \Rightarrow N_{p} = \frac{ρ {gQ}_{p} H_{p}}{η}$ (6)

When the water supply index is given, the optimal flow (Q_B) corresponding to the maximum efficiency under the head can be obtained from laws of similitude and constant efficiency value curve as: $Q_{B} = \sqrt{\frac{H_{P}}{H_{A}}} Q_{A}$ (7)

When M pumps of the same type are opened in parallel, the total power is shown in Equation (8): $P_{total} = γ R_{0} Q_{0} \sum_{i = 0}^{M} \frac{Q_{i}}{c_{1} \frac{Q_{i}}{S_{i}} + c_{2} \frac{Q_{i}^{2}}{S_{i}^{2}} + c_{3} \frac{Q_{i}^{3}}{S_{i}^{3}}}$ (8) $Q_{0} = Q_{1} + Q_{2} + . . .$ (9) $S_{i} = Q_{i} / Q_{N}$ (10)

where, P_total is the total power, γ is the water weight and it is a fixed value, R₀ is the drag coefficient of the system pipe network, Q₀ is the water demand, S_i is the frequency conversion ratio of pump i.

When multiple pumps are connected in parallel, only the output water flow and frequency conversion ratio of each pump are the same case the most energy saving [29 –31]. Through the derivation of Equation (8), it represents the total power of multiple pumps in parallel, when Q₁ = Q₂ =… , S₁ = S₂ = …, the total power of the pump group is the smallest. According to Equation (7), the optimal pump water flow rate under the given water supply H_p index head is Q_B. When the total number reaches m_temp = Q₀/Q_B, the energy consumption of pump sets is the lowest. But the m_temp probably not be an integer. At this time, the optimal number of pumps that are turned on should approach that integer, Equation (11) represents the function of rounding down. $m = [m_{temp} + 0.5]$ (11)

Substitute into the quadratic Equation (5) to solve, S is denoted by: $S = - \frac{b_{1} Q_{0}}{2 b_{0} m} \pm \sqrt{\frac{b_{1}^{2} Q_{0}^{2}}{4 b_{0}^{2} m^{2}} - \frac{b_{2} Q_{0}^{2} - m^{2} H_{p}}{b_{0} m^{2}}}$ (12)

generally, to meet the demand for water supply, Q₀ is a constant in a certain period of time. Equation (12) shows that S is a function of H_p. When the end water flow is constant, the total head of the pump combination can meet the requirements within a certain range.

3 Mathematical modeling and constraints

3.1 Mathematical modeling

Based on the above problem statement and cost component analysis, considering the flow-power performance curve of the pump in parallel, the following optimal scheduling model is established with the goal of reducing operating energy consumption, the number of pump switches and load balancing.

Minimum total power consumption of the pump group

Because the change of water production speed and water demand data of the water plant leads to the dynamic change of the water surface height of the savings pool, the energy consumption required to pump the same amount of water from the savings pool to the municipal pipe network is different. Therefore, considering the influence of the water surface height change, the electricity consumption of pumping includes the energy consumption of the pump and the energy consumption of lifting water from the pool to the water supply pump station, which is calculated by Equation (13). $min C = \sum_{t = 1}^{T} e_{t} [\sum_{i = 1}^{M} w_{ti} P_{ti} + {kQ}_{t - all} (h - h_{t})]$ (13)

Where, $\sum_{i = 1}^{M} w_{ti} P_{ti}$ is the power consumption of all the pumps, and kQ_t-all (h - h_t) is power consumption used to increase potential energy. T is the time period of scheduling program (usually 24 hours). e_t is the price of electricity at time t. M is the number of all pumps in the water supply station. w_ti is decision variable (0 or 1). P_ti is the power of pump i, the specific calculation formula is as Equation (14). $P_{ti} = c_{1} S_{ti}^{3} + c_{2} S_{ti}^{2} Q_{ti} + c_{3} S_{ti} Q_{ti}^{2}$ (14)

Where, S_ti and Q_ti are the frequency conversion and the water flow of pump i during time t. k is the amount of energy needed to lift one meter per cubic meter of water. Q_t-all is total water amount during time t, h and h_t are the heights of both the pump and the water surface relative to the bottom of the pool during time t.

Minimum the number of pump switches

In addition to reducing the cost of electricity, it is also necessary to optimize the maintenance cost of the pump. Frequent starting or opening of the pump will not only damage the relevant parts of the pump, but also cause obvious fluctuations in the pipe network pressure [32, 33]. Therefore, the number of pump switches can be regarded as an indicator of maintenance costs and reducing pump switching times can effectively reduce maintenance costs, which is calculated by Equation (15). $\min R = \sum_{t = 1}^{T - 1} \sum_{i = 1}^{M} (| R_{(t + 1, i)} - R_{(t, i)} |)$ (15)

Where, R is the total number of pump switches, and R_(t,i) represents the status variable of pump i during time t, which takes the value 0 or 1.

Minimum the load balancing

In the actual pump scheduling of water plant, some pumps work continuously and some pumps are idle for a long time, that leads to frequent replacement or maintenance of the pump. Therefore, the third objective function is introduced to achieve load balancing. Balancing the service time of each pump can help to reduce operation and maintenance costs. It is calculated by Equation (16). $min L = \sqrt{\frac{\sum_{i = 1}^{M} (X_{i} - \bar{X})^{2}}{M}}$ (16)

Where, X_i is the total working time of pump i, and $\bar{X}$ is the mean value of each pump working time.

3.2 Constraints

In general, the constraints of pump scheduling problems are related to water flow, head, frequency conversion ratio, efficiency, etc. The following equations present the constraints to define the feasible solutions: $Q_{t - all} = \sum_{i = 1}^{M} Q_{ti}$ (17) $Q_{i, min} ⩽ Q_{i} ⩽ Q_{i, max}$ (18) $Q_{pret, j} ⩽ max Q_{tj}$ (19) $Q_{t - all} < Q_{pool, t}$ (20)

Equation (17) indicates that the total water flow (Q_t-all) of the water supply pump station at any time in the cycle is equal to the sum of the water output of each pump (Q_ti); Equation (18) indicates that the water flow (Q_i) of the pump should be in the set range (Q_i,min to Q_i,max) when it works, and the constrain is to ensure the safety and reliability of the pump and meet the water supply index; Equation (19) indicates that the water demand (Q_pret,j) in each area at any time is less than the maximum water flow(Q_tj) that the area can provide; Equation (20) indicates the total water demand (Q_t-all) at any time is less than the total amount of water (Q_pool,t) in the savings pool. $| H_{jt} - H_{pre, jt} | ⩽ σ$ (21) $H_{i, min} ⩽ H_{p} ⩽ H_{i, max}$ (22)

Equation (21) indicates that the deviation between the actual water pressure value (H_jt) and the predicted pressure value (H_pre,jt) in each area at any time period is within a certain range (σ); Equation (22) indicates that the pressure value (H_p) of each pump in the stable operation stage is within the allowable range (H_i,min to H_i,max) when it is turned on. $S_{i, min} ⩽ S_{ti} ⩽ S_{i, max}$ (23) $n_{i, min} ⩽ n_{ti} ⩽ n_{i, max}$ (24)

Equations (24) respectively represent the frequency conversion ratio (S_ti) and the speed (n_ti) setting range (S_i,min to S_i,max, n_i,min to n_i,max) of pump to ensure more efficient operation. $β_{min} ⩽ | \frac{η - η_{max}}{η_{max}} | ⩽ β_{max}$ (25)

Equation (26) indicates that the operation efficiency(η) of the pump shall not be lower than a certain value, and η_max is the maximum efficiency value determined by theoretical research, β_min and β_max is the set threshold value. $\sum_{c = 1}^{C_{j}} w_{c, j} > 0$ (26)

To ensure that there are open dispatch pumps allocated to the area at any time, Equation (26) indicates that the sum of the decision variables (w_c,j, is 0 or 1) of the pipeline network in each water supply area corresponding to the pump at any time, and C_j is the total number of pumps in area j. $T_{i} ⩽ T_{max}$ (27)

To avoid damage to the motor due to long-term use, Equation (27) indicates that the continuous working time (T_i) of any pump shall not exceed the maximum value (T_max).

4 The proposed method

The most obvious feature of the pump groups optimization model is that the water pressure and demand at the pipe network end change with time. Therefore, it is necessary to obtain the optimal frequency conversion ratio and the number of pumps started at each moment based on the known water demand, and then adjust the pump switch situation in a day. The specific solution process is shown in the following sections.

4.1 The solution steps based on genetic algorithm

The flow chart of the pump scheduling problem solved by the improved genetic algorithm is shown in Fig. 4. The specific steps are as follows:

Fig. 4

Solution flow chart based on genetic algorithm.

Step 1: Set experimental parameters, and stores the predicted water demand value of each area in the period in a two-dimensional array;

Step 2: Fit the flow-efficiency and flow-pressure curves at the rated speed according to the experimental data, determine the water supply index corresponding to the maximum efficiency, and calculate according to the water demand index of each period, the water production data of the water plant, and the water volume data of the reservoir and the change curve of the water surface height of the clear water pool in the cycle;

Step 3: Set the relevant parameters of the first objective function, including the setting of the constraint condition parameters, the value range of the water supply pressure in each area, etc.

Step 4: Obtain an initial population of size n by random generation, and set the maximum number of iterations to genMax;

Step 5: Calculation of the fitness value of the corresponding pipe network of a single chromosome is made. After selection, a new population is obtained by recombining the chromosomes, and then the optimal speed regulation ratio and the number of opening pumps of the variable frequency pump of each regional pipe network is got through crossover and mutation.

Step 6: Repeat Steps 3, 4, and 5, and store the obtained combination of the optimal pump opening numbers corresponding to each time period into a J*24 two-dimensional array, and record frequency conversion ratio data, head data, water flow data and so on;

Step 7: Randomly generate n two-dimensional chromosome group within a reasonable population range according to the chapter 4.3 initialization population method, set the maximum number of iterations genMax, and obtain a new population through selection, crossover, and mutation. Finally, the optimal individual is obtained by iterative algorithm. The load balancing value of the total running time of each pump is counted, and the individual with the smallest load balancing value is obtained by iteratively performing the above operations, which is the optimal opening and closing scheme for pump group scheduling;

Step 8: The termination condition is set to the number of iterations, and the scheduling scheme is changed according to the actual situation. For example, the pump will be in a deactivated state such as maintenance and maintenance for a certain period of time in the future. The position of the disabled state in the two-dimensional array is set and the value in the two-dimensional array is set to 2, and it does not participate in crossover and mutation;

Step 9: Output the optimal scheduling scheme and its related data, and the algorithm ends.

4.2 Individual encoding and decoding

As shown in Fig. 5, binary coding [34] is used for each area and 0 and 1 sequences are randomly generated. The length of the chromosome is set as J*L, in which J represents area numbers, and the decoding is converted into the corresponding value according to the set pump head value range. The decoding process as shown in the figure below obtains the value of the current gene coding area. When the binary code corresponding to each area of the chromosome is converted into the corresponding decimal head within range, the setting range of the head size is obtained based on experimental data and historical experience.

Fig. 5

One-dimensional chromosome coding and decoding method.

The data obtained in the first stage is applied to the stage 2, and then the binary coding is also used. As shown in Fig. 6, the chromosome adopts the two-dimensional chromosome coding method. The length of the chromosome is the size of the period value T, and the width is total number of pumps of each area, the meaning of the chromosomes is: 0 means the pump is off, 1 means the pump is on. According to the predicted data before scheduling and the actual operating status of the pump, there will be a third situation. If the pump is in a state of failure, repair, replacement, etc., at some time, the code is denoted by 2.

Fig. 6

Two-dimensional coding method of objective function.

In most water purification plants, the pump group control method based on time priority is adopted, which means when pump is added next moment which will select the pump that shortest total historical running time turned on, and when the number of pumps turned on is reduced at the next moment, the pump with the largest total historical operating time will be turned off. The third objective function in this paper is performed after the second objective is optimized. Only need After repeated iterations of the second objective function, results are compared to obtain a scheduling scheme with less load balancing.

4.3 Population initialization and fitness value calculation

The initialization of the population is a key operation in the genetic algorithm [35 –37]. For the first optimization objective, a chromosomal gene code is randomly generated composing of a binary string with a population size of N and length of J*L, where in each segment in each chromosome is of length L, which represents the gene code of each area and is independent of each other. For the second and third optimization objectives, the two-dimensional chromosome individual is composed of multiple column vectors, where each column vector represents a different time period, and the generated 0, 1 sequence mainly depends on each of the obtained optimal number of pumps that are turned on in each area in the first optimization objective. Since there are flexible pumps in each area, the flexible pumps only supply water to one area anytime, so it is necessary to additionally judge whether the scheduling requirements of the flexible pumps are met.

For the calculation of fitness value, according to the decoding method, frequency conversion ratio and water supply flow value were obtained by plugging the pressure value into the fitting curve. Because the objective function is to find the minimum value, the larger the value of the objective function, the lower the fitness is. The fitness value of each individual in the population of each area is calculated according to the fitness function $f (S) = - m \times c_{1} S^{3} + c_{2} S^{2} Q_{ti} + c_{3} {SQ}_{ti}^{2}$ for subsequent genetic operations, and m is the number of pumps to be turned on for the area, where Q_ti = S · Q_N. Similarly, for the second and third objectives, the larger the objective function result, the smaller the fitness value.

4.4 Genetic algorithm optimization process

In the process of using genetic algorithm to solve the initialized population, it generally includes operations such as selection, crossover, mutation, etc. [38]. After iterative cycle, the adaptability of general offspring individuals is greater than that of the parent, so as to obtain the optimal individual.

1) Selection

The selection adopts the championship selection strategy, as shown in Fig. 7. The specific steps are to select a group of individuals from the initialized population each time, and select the best individual to enter the offspring population according to the fitness value of the individual. Selected the position of the individual parent is randomly regenerated and replaced, and this operation is repeated until the size of the child population reaches the size of the parent population and enters the next cycle.

Fig. 7

Champion selection strategy.

2) Crossover

As shown in Fig. 8, when solving the minimum energy consumption, partial crossover is adopted, and two points are randomly intercepted from the two parent chromosomes, and the genes in the points are exchanged to obtain new offspring individuals, and their fitness is calculated. If the fitness value of the offspring is large, it will enter the next generation population.

Fig. 8

One-dimensional chromosomal chiasma.

As shown in Fig. 9, when the two-dimensional array chromosome is crossed, the number of pumps on each time period and each area calculated by the first objective is fixed because the encoded value indicates the on-off state. It means that the number of 0 and 1 distribution in each area of the chromosome column is determined, so in the process of chromosome crossover, two parent chromosomes are randomly selected, and in a certain water supply area in a certain column of parent chromosome 1 randomly select n genetic loco with 0 and 1, and then extract n loci from the parent chromosome 2. The 0 and 1 numbers of the loci extracted by the two parents must be the same, otherwise they will be randomly selected again, exchange in the original order, and then determine whether it meets the scheduling conditions of the flexible pump, and compare the fitness values of the parent and the child. If the fitness value of the child becomes larger or unchanged, it will enter the next generation.

Fig. 9

Two-dimensional chromosomal chiasma crossover.

3) Mutation

As shown in Fig. 10, in (a), for the chromosomal mutation in the first optimization objective, a mutation point is randomly generated on the locus, and its value changes from 0 to1or 1to 0; for a certain area in one column of a two-dimensional chromosome, a gene mutation site is randomly generated to convert its gene position 1 (0) to 0 (1). Because of the constraint of the number of pump groups in a single column, it is necessary to balance the number of pump groups, which means the gene locus containing the value of 0 is mutated again. If the mutated gene locus belongs to the flexible section, the coding of the pump in the regional locus should be modified at the same time, and judge whether it meets the flexible pump scheduling conditions, and generate offspring. Whether to enter the next generation by comparing with the fitness value of the parent.

Fig. 10

Chromosome mutation.

5 Case study

All experiments are implemented by using MATLAB2018a and IDEA2021, which are run on Windows10 x64-bit system with Intel Core i5 9400F- 2.90 GHz CPU and 16.0 GB of RAM.

5.1 Description of test data

In order to verify the feasibility of the objective function model and the solution algorithm, this study selected the water demand data of a water plant in Suzhou, China on one day in 2019, which supplies water to three areas. The model of pump is RDL V500-640A. The performance curve is fitted as shown in Fig. 11(a) and 11(b). When the pump has the highest efficiency, Q_A is 1.068m3/s and H_A is 45.77 m. The reservoir depth of the water plant is 4 meters, k (the amount of energy needed to lift one meter per cubic meter of water) is equal to 0.0032. The electricity consumption of the water purification plant is affected by time-of-use electricity prices for industrial use, including peak hours, normal hours and valley hours. For 24 hours a day, the peak hours include 8 : 00–12 : 00 and 17 : 00–21 : 00, the normal period is 12 : 00–17 : 00 and 21 : 00–24 : 00, and the valley period is 0 : 00–8 : 00. And the electricity prices are 1.025 yuan/KW · H, 0.725 yuan/KW · H and 0.425 yuan/KW · H respectively.

Fig. 11

Data graphs for case study.

The experiment is based on the study that the water supply pump station of the water purification plant supplies water to 3 area pipeline networks at the same time, the water demand data of each area within 24 hours of one day is drawn as Fig. 11(c) and Table 2. And Fig. 11 (d) describes the variation curve of water depth in the pool.

Table 2

Water demand in each area

Time (h)	Area 1	Area 2	Area 3
0	4282	6579	6949
1	2506	4357	5497
2	2307	3854	5049
3	2239	4691	4987
4	3456	4067	6087
5	5382	6978	8143
6	6971	8354	9864
7	9976	13572	14687
8	12690	14530	13246
9	9160	11874	15793
10	9059	9383	17547
11	8762	10694	19230
12	9520	12348	21560
13	9862	14369	17593
14	8267	16759	15698
15	7561	11654	12841
16	7751	9357	16871
17	9563	8694	19871
18	9957	13564	18667
19	11830	12364	18357
20	13357	13789	16987
21	11058	17689	16794
22	9853	12689	12367
23	6958	8697	9874

Because the distance between the municipal pipe network and the water purification plant in each area is different, and the water supply pressure in each area is not the same, so the set range values are shown in Table 3.

Table 3

Pump head range of each area

	Area 1	Area 2	Area 3
Minimum Pump Head (m)	31.2	35.0	35.4
Maximum Pump Head (m)	35.3	38.1	37.2

As shown in Table 4, there are 14 pumps in the water supply station which are numbered as No. 1-14. Among them, there are 10 common pumps and 4 flexible pumps. The flexible pump is No. 4, No. 8, No. 9 and No. 14. The common pump only distributes water to one corresponding fixed area, while the flexible variable frequency pump can deliver water to multiple areas. In this paper, the additional energy consumption such as switching valves brought by the flexible pump is ignored, and energy consumption and water supply indicators is calculated according to the normal variable frequency pump.

Table 4

Pump number and distribution of each area

	Common pump	Flexible pump
Area 1	1, 2, 3	4, 9
Area 2	5, 6, 7	8, 9, 14
Area 3	10, 11, 12, 13	4, 8, 14

5.2 Result presentation

First, power consumption of the optimal pump group in each area and each time period is solved, then parameters in the genetic algorithm is set: the population size NP is 100, the chromosome length L is 3*30, the crossover probability Pc is 0.8, the mutation probability Pm is 0.15, the number of iterations of the main loop is 50 times, and the total water flow value and head range of each area are substituted. To simplify the calculation, the minimum cost of power consumption per hour for each area is expressed as total power. The calculation results of the first stage are shown in Fig. 12 and Table 5. It can be seen that the water outlet pressure in each area is relatively stable and the efficiency is high.

Fig. 12

Power consumption and the number of pumps that need to be turned on in each area.

Table 5

Electricity consumption of each Area

Time(h)	Area 1			Area 2			Area 3
	S	P(KW ·h)	Number	S	P(KW ·h)	Number	S	P(KW ·h)	Number
0	0.91	471.27	1	0.87	765.49	2	0.89	818.04	2
1	0.79	266.60	1	0.95	525.91	1	0.84	654.95	2
2	0.78	249.10	1	0.91	451.34	1	1.01	648.22	2
3	0.77	242.19	1	0.98	575.97	1	1.00	633.47	1
4	0.85	356.48	1	0.93	475.88	1	0.86	706.58	2
5	0.80	550.70	2	0.88	790.82	2	0.93	951.82	2
6	0.85	697.39	2	0.94	956.87	2	0.87	1109.54	3
7	0.84	987.37	3	0.88	1501.58	4	0.90	1651.25	4
8	0.83	1257.54	4	0.89	1619.21	4	0.88	1485.67	4
9	0.82	919.47	3	0.85	1339.21	4	0.87	1793.62	5
10	0.82	929.87	3	0.86	1075.16	3	0.89	2032.97	5
11	0.81	906.32	3	0.89	1234.22	3	0.87	2238.07	6
12	0.83	984.54	3	0.86	1425.82	4	0.89	2519.13	6
13	0.83	1019.59	3	0.89	1659.38	4	0.89	2049.86	5
14	0.80	859.33	3	0.87	1928.76	5	0.86	1828.58	5
15	0.87	790.55	2	0.91	1351.62	3	0.87	1486.20	4
16	0.88	807.28	2	0.86	1063.35	3	0.88	1935.71	5
17	0.83	969.18	3	0.85	991.47	3	0.87	2273.24	6
18	0.84	1009.26	3	0.88	1533.22	4	0.86	2136.69	6
19	0.81	1200.14	4	0.86	1401.84	4	0.90	2110.84	5
20	0.84	1360.85	4	0.88	1566.18	4	0.88	1949.28	5
21	0.86	1150.76	3	0.89	2029.08	5	0.88	1942.87	5
22	0.83	1023.34	3	0.86	1467.31	4	0.86	1447.79	4
23	0.85	729.34	2	0.85	1016.82	3	0.87	1158.05	3
Mean value	0.829	822.47	2.50	0.886	1197.77	3.08	0.889	1565.10	4.04

Then, substitute the results of the number of pumps in each area shown in the second step into the improved genetic algorithm. The number of iterations set by the algorithm genMax is 50. The results obtained by the algorithm after many experiments are all in the number of opening and closing converge at 29. Set the number of iterations of the third objective function to 50. Figure 13 shows the iterative convergence diagram of the load balancing and the number of pump switches.

Fig. 13

Convergence diagram of load balancing and the number of pump switches.

Table 6 shows the on-off status of each area and each pump within 24 hours, 0 means off (1 means on), the shaded area is the area to which the flexible pump belongs, and the corresponding optimal scheduling Gantt chart is shown in Fig. 14. The ordinate in the figure is the number of each pump, and the intersecting part with color means that the pump supplies water to different areas without participating in the opening and closing process. The water supply area is changed by switching the valve, and the last line is the calculated value of load balancing.

Table 6

Pump scheduling information with time

Time (h)		0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
Area 1	M₁	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	1	1	1	1	1	1	1
	M₂	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0	0
	M₃	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
	M₄	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	1	1	0	1	1	1	1	0	0
	M₉	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	1	1	0	1	0
Area 2	M₅	1	0	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1
	M₆	1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	1	1	1	1	1	1	0
	M₇	0	0	0	0	0	0	0	1	1	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1
	M₈	0	0	0	0	0	0	0	0	1	0	0	0	0	1	1	1	0	0	0	1	1	1	0	1
	M₉	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	1	0	1	0	0	1	0	0
	M₁₄	0	0	0	0	0	0	0	1	0	1	1	0	1	1	1	0	0	0	0	0	0	0	1	0
Area 3	M₁₀	1	1	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1
	M₁₁	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0
	M₁₂	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0
	M₁₃	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0	1	1	1	1	0	0
	M₄	0	0	0	0	0	0	0	1	0	1	1	1	1	1	1	0	0	1	0	0	0	0	1	1
	M₈	0	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0	1	1	1	0	0	0	1	0
	M₁₄	0	0	0	0	0	0	0	0	1	0	0	1	0	0	0	1	1	1	1	1	1	1	0	1

Fig. 14

Gantt chart of pump scheduling.

5.3 Sensitive analysis

In order to be more in line with the real situation, disturbance factors are also added to assess the effects under different pump operating conditions in this paper.

5.3.1 Pump scheduling considering common pump failures (M1)

Assuming that the pump M₅ needs to be maintained between 11–14 o’clock. The running result is shown in Fig. 15, where the red area is inactive. Even if when the pump is faulty, the pump scheduling can still run reliably.

Fig. 15

Gantt chart of pump scheduling.

5.3.2 Pump scheduling considering flexible pump failures (M2)

When the flexible pump in the water supply pump room fails or needs maintenance, more common pumps need to be added to meet the demand. At this time, all pumps in the pump room are common pumps. The running result is shown in Fig. 16.

Fig. 16

Gantt chart of pump scheduling.

5.3.3 Comparison of experiment results

As shown in Table 7, when a common pump in a water supply pump station breaks down and needs to be repaired, although the load balancing is increased, the normal water supply work of the pump station is guaranteed and the number of pump switches remains unchanged. When the flexible pump in the water supply pump station needs maintenance, the number of pumps in service increased by 4, the number of pump switches is relatively reduced by 6, and the load balancing is reduced by 50.31%.

Table 7
Comparison of experiment results

Number of Number of Load

pumps required pump switches balancing

The proposed model 14 29 1.239

M1 14 29 1.880

M2 18 35 2.291

	Number of	Number of	Load
The proposed model	14	29	1.239
M1	14	29	1.880
M2	18	35	2.291

5.4 Performance analysis of algorithms

The software CPLEX 12.9.0 is used to verify the correctness of the proposed model. Since the true solution cannot be obtained within 2 hours for solving large-scale problems, 5 sets of small-scale experiments are generated for model validation. For example, instance 2-8-12 represents 2 water supply areas, 8 pumps, and 12 hours of dispatch time. Figure 17 shows the comparison result of CPLEX and IGA. The result shows that the true solution obtained by CPLEX verifies the correctness of the proposed algorithm.

Fig. 17

Comparison results of CPLEX and IGA.

In order to verify the performance of the proposed algorithm in pump scheduling, particle swarm optimization algorithm (PSO) and simulated annealing algorithm (SA) are used as the comparison algorithms. In PSO, the number of particles is 100, the maximum number of iterations is 50, the learning parameter is 3 and the maximum speed is 0.1. In SA, the initial temperature is 100, the lower bound of temperature is 1, the maximum number of iterations is 50, and the temperature drop rate is 0.98. The two algorithms are tested for 10 times respectively, and the experimental results are shown in Table 8. Compared with PSO and SA, the electricity cost (F1) obtained by IGA is 0.421% and 0.346% less, respectively. And the pump switch number (F2) obtained by the proposed algorithm is 2 less than that of the comparison algorithm. What’s more, the goal of load balancing (F3) is lower than the other two algorithms. It means that the comprehensive performance of the proposed algorithm is superior to other algorithms. Although the average running time (ART) of IGA is longer than that of the comparison algorithms, the solution obtained by IGA outperform the other two algorithms.

Table 8

Comparison results of different algorithms

	F1	F2	F3	ART (S)
IGA	67872.80	29	1.239	194.4
PSO	68158.78	31	3.134	98.0
SA	68107.87	31	4.338	176.0

6 Conclusion

Aiming at the problems existing in the pump scheduling system of the current water purification plant, three factors affecting pump scheduling cost are put forward. It mainly includes the electricity cost of pumping water and the pump maintenance cost, and the pump maintenance cost is represented by the number of pump switches and load balancing. After the deep exploration of the performance parameters of the parallel pump set, the mathematical model of the minimum energy consumption, the minimum number of pump switches and the minimum load balancing is established. The effects of water depth, parallel pump performance, pump duration and so on are considered in the mathematical model and constraints. In order to solve this problem, a two-stage strategy based on genetic algorithm is proposed. Firstly, the frequency conversion ratio was set as the feasible solution variable in the genetic algorithm for coding, and the lowest electricity cost of the pump and the number of pumps need to be turned on were obtained through genetic operation. Then, the number of pumps need to be turned on in the first step was substituted into the second step as input to initialize the two-dimensional chromosome population, and the optimal number of pump switches and load balancing were obtained based on genetic algorithm and iteration. Then, considering the impact of emergencies in the scheduling process on the scheduling plan, the improved genetic algorithm is presented to get a on and off state scheduling scheme of each pump in each period of the day in a fast and efficient way. Finally, a case study from a real waterworks in Suzhou, China is used to verify the validity of the proposed model. This paper provides guidance for the pump scheduling tasks of water purification plants in the new era.

In this paper, the pumps in the water supply pump station are all of the same model and the water demand data of the pipe network is known. In the future study, the parallel operation of different models of pumps will be considered and the pump scheduling will be carried out according to real-time data.

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Optimization of pump scheduling in waterworks considering load balancing using improved genetic algorithm

Abstract

Keywords

1 Introduction

2.1 Problem statement

3.1 Mathematical modeling

4.1 The solution steps based on genetic algorithm

4.4 Genetic algorithm optimization process

5.1 Description of test data

5.3.1 Pump scheduling considering common pump failures (M1)

Table 7 Comparison of experiment results Number of Number of Load pumps required pump switches balancing The proposed model 14 29 1.239 M1 14 29 1.880 M2 18 35 2.291

References

Table 7
Comparison of experiment results

Number of Number of Load

pumps required pump switches balancing

The proposed model 14 29 1.239

M1 14 29 1.880

M2 18 35 2.291