Selection of water pump combination schemes and their optimization procedures

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I. Introduction A pumping station is a structure in the water supply system used to maintain water pressure, lifting the required amount of water to the desired height. Overall, in the water supply system, the water distribution system, which consists of pumping stations, water conveyance pipes, pipe networks, and regulating structures, is the subsystem with the highest investment. In the selection of water pumps for pipeline network design, a technical and economic comparison of various options is often carried out within the range of possible variations in flow rate and head. It should be based on the technical feasibility of using this solution within the design life span, that is, whether it can meet the requirements for various water usage scenarios. The optimization of pump combinations and their characteristics is an important aspect of the design of water supply pump stations (pump houses), and it holds significant importance for the technical and economic benefits of such stations after they are built. The operational energy consumption in water supply systems is primarily due to the energy use of various pump sets; therefore, the optimal design of pump combinations essentially aims to minimize energy consumption while meeting the requirements for water flow rate and pressure. 　　The energy savings of variable-speed water pumps are mainly reflected in two aspects: (1) After the speed of the pump is adjusted, the operating point of the pump changes, resulting in an increase in its efficiency. (2) Under the condition of meeting service pressure requirements, the water pump reduces its water supply pressure through speed regulation, thereby decreasing power consumption. 　　With the continuous expansion of water supply scale, variable-speed pumps have become widely used in water supply systems. In current designs, a combination of variable-speed and constant-speed water pumps is often used to reduce overall energy consumption. This article discusses how to select a pump system for different flow rates and head values, and how to adjust the speed of pumps when such a system includes variable-speed pumps, in order to change the operating point of the water pump unit and thereby improve its efficiency. The program in this article is implemented using advanced assembly language and VC++6.0. II. Calculation Methods and Working Principles 2.1 Curve Fitting Curve fitting is a very practical method, and especially in the era of rapid development of modern computers, the issue of fitting becomes even more important. Curve fitting is approached in the following way: suppose we have a set of experimental data that appear chaotic, denoted as (*, yi), where i = 1, 2, …, m*n. We wish to identify patterns within this data; in other words, we want to construct an approximate function s(x) that can approximate the desired function y = f(x). Linear least squares is more commonly used in practical applications. The least squares solution is: s*(X) = a0*φ0 + a1*φ1 + Λ + ax*φx. The coefficients a0*, a1*, Λ, ax* of the least squares solution can be obtained by solving the system of equations. 　　As a common case of curve fitting, when algebraic polynomial fitting is considered, that is, when {φ0,φ1, …,φn} = {1, x, x2, …, xn}, then the corresponding normal equations are as follows. Here, ωi = ω(*) represents the weight function, which indicates the importance of different points (*, yi). For example, the weight ω(*) at a point (*, yi) can be used to represent the number of times that data point appears in the experiment, or it can be used to indicate the accuracy of yi; the more accurate yi is, the more important its role is, and consequently, the larger the weight ω(*) becomes. 　　Abbreviate “absMiddle>” as “∑”. 　　2.2 Variable-speed operation Variable-speed operation refers to the operation of a water pump driven by an adjustable-speed motor, whereby the operating point of the water pump system is changed by altering its speed. For constant-speed operation, the focus is on how to make full use of the high-efficiency operating \"region\" on the Q–H curve of the centrifugal pump under fixed, single rotational speeds. Therefore, for the variable-speed operation mode, attention will be paid to how to make full use of the highly efficient operating \"region\" of the centrifugal pump’s Q–H curve resulting from speed variation, in light of the hourly variations in water consumption in urban water supply networks. Therefore, operating at a variable speed **significantly expands the effective operating range of centrifugal pumps, making it a very reasonable method of adjustment in pump station operation. 　　By applying the law of similarity to the same vane pump operating at different rotational speeds, the following equations can be obtained. These three equations indicate that for the same vane pump, as the rotational speed n changes, other performance parameters will change according to the aforementioned proportional relationships. The above three equations represent a special form of the law of similarity, known as the law of proportionality. For users of water pumps, the law of proportionality is very useful. It reflects the pattern of changes in the main performance of the water pump as the rotational speed changes. III. Implementation of the Program The process for selecting a pump set involves the following steps: The program first uses the excellent interface capabilities of VC++6.0 to plot the characteristic curves of the water pump. Next, determine the operation plan. If there are two pumps, then there are three operating modes: ① Pump 1 # ② Pump 2 # ③ Pump 1 # and Pump 2 # in parallel. For the parallel connection of pumps, the author adopts the principle of flow addition at equal head. Curve fitting is performed on the resulting combined flow rate and head values to determine the curve equation. This program allows the number of points for curve fitting to be entered manually via a dialog box. 　　Data (flow rate, head) can be entered using the interface dialog box, or by simply clicking the left mouse button. If the selected scheme does not support speed control, the interface will display the selected scheme number along with its Q—H curve. If the pump in the scheme can be adjusted in speed, first adjust the speed of the pump with variable speed control, and then combine the pumps included in the scheme and perform curve fitting to obtain the characteristic curve of the pump after speed adjustment. At the same time, the selected water pump combination scheme and the combination curve equation will be displayed on the interface. IV. Examples The following analyzes the matter through specific examples. 　　A pumping station is equipped with three water pumps:
1# – Characteristic curve: H=68-0.0000029Q2; efficient operation range: 2900–3300 L/s.
2# – Characteristic curve: H=72-0.000012Q2; efficient operation range: 1250–1600 L/s.
3# – Characteristic curve: H=70-0.0000022Q2; efficient operation range: 3400–3800 L/s.

Operation scenarios:
① 1#
② 2#
③ 3#
④ 1# + 2#
⑤ 1# + 3#
⑥ 2# + 3#
⑦ 1# + 2# + 3#

Pump 3 is a variable-speed pump, and scenarios ③, ⑤, ⑥, and ⑦ all involve the use of this variable-speed pump. H 1# 2# 1#+2# H1 Q1 Q2 Q1+Q2 　　For a given H value, the corresponding flow rates are determined using the characteristic curves for 1# and 2#, and then these values are added together. For other options, the same logic applies. 　　Curve fitting is performed using least squares estimation: H0 and S0 can be determined based on equation (3). The results for this example are as follows: H4 = 69 – 0.00000130Q2, H5 = 69 – 0.00000063Q2; H6 = 70 – 0.00000108Q2, H7 = 69 – 0.00000042Q2. To ensure that the characteristic curve of the pump system after speed adjustment passes through the given input point, it is first necessary to determine the flow rate and head (H2, Q2) corresponding to the speed-adjusted pump when it is not connected in parallel. This is done using the similarity theorem and the principle of equal efficiency (the H/Q2 ratio is constant at all points where the efficiency is equal, and this constant is denoted as k). Using this k value, a parabola with equal efficiency and similar operating conditions can be plotted. ） 　　H1/Q12=k=H2/Q22 ........（4） 　　H1=H0-S0Q12......................（5） 　　Using equations (4) and (5), the values of H1 and Q1 can be determined. Since 　　Q1/Q2=n1/n2, it follows that H1/H2=(n1/n2)2. 　　For any given value of H1 and Q1 on the original curve, there correspondingly exist values of H2 and Q2. By fitting the resulting point value curve, the characteristic curve equation after speed regulation is obtained. Then, by superimposing the water pumps from the pump set options, the characteristic curve of the selected option can be obtained. V. Conclusion 　　Due to the high energy consumption of pump stations in water supply systems. Therefore, achieving the goal of minimizing pumping station costs will bring significant social and economic benefits. This program was developed on a computer using VC++6.0, and it can be useful for selecting pump system configurations. 　　6. Marshall Brain, Lance Lovette. MFC Developer’s Guide. Machinery Industry Press, 1999