Economic Operation of Variable Speed and Blade Angle-Adjustable Pumping Stations of an Open-Channel Water Transfer Project

Abstract

A large amount of energy would be consumed for open-channel water transfer projects due to the low efficiency of pumping stations. One measure to improve the efficiency of a pumping station is to install variable-frequency drives (VFDs). In this paper, a discharge optimization model is proposed for a single pumping station equipped with different numbers of variable speed and blade angle-adjustable pump (VSBAP) units, and then a head optimization model is proposed for cascade pumping stations. The study on the Miyun Reservoir Regulation and Storage Project in China shows that the installation of VFDs can increase the number of operable conditions of a single BAP unit by changing the blade angle and speed and ensure the high efficiency of the pumping unit under most operating conditions, thus reducing the energy consumption of the pumping station. It is desirable to install two VFDs in the Tuancheng Lake–Huairou Reservoir section to ensure the long-term operation of the cascade pumping stations in an economically profitable way. In conclusion, the installation of VFDs can effectively reduce the operation cost of cascade pumping stations.

1. Introduction

Water transfer projects are designed to alleviate the supply–demand imbalance of water resources [1]. Pumping stations are responsible for lifting water from low to high altitudes [2], and thus they form an integral part of most water transfer projects, such as the East Route Project of the South-to-North Water Transfer Project, the Yangtze-to-Huaihe Water Transfer Project, and the Miyun Reservoir Regulation and Storage Project in China [3,4,5,6]. It is notable that pumping accounts for a large fraction of the electricity consumption of a water transfer project [7,8,9,10]. The ever-increasing water demand leads to the operation of more pumping stations for longer periods of time and, consequently, higher electricity consumption [11], and what makes the problem more acute is the increase in electricity prices, which can substantially increase the pumping cost [12,13]. Therefore, it is necessary to operate these pumping stations in an energy-saving manner in order to maximize the economic benefit of the water transfer project. Some attempts have been made to optimize the operation of water transfer projects consisting of a single pumping station [14,15] or cascade pumping stations [16]. The objective functions of previous optimization models are often set to minimize energy consumption [13,17,18] or operating costs [19] or maximize operating efficiency [16,20]. These optimization problems are often solved using dynamic programming [21], integer programming [22], genetic algorithms [23], and particle swarm optimization [24]. Many hybrid algorithms, such as the extended dynamic programming algorithm (EDPA) [25], the ant colony optimization algorithm (ACO), and the simplex method (SM) [26], have also been proposed to improve the convergence speed of the model and avoid dimensional disaster in solving complex problems.

A common approach to optimizing the operation of a single pumping station is to obtain the on–off state and operating discharge of each pump group so that its highest operating efficiency or lowest energy consumption/operating cost for a given head can be obtained. Low-head large-discharge pumps are generally used in open-channel water transfer systems, among which the axial-discharge pump is the most common choice. Ghassemi-Tari et al. [27] transformed the discharge optimization problem of the pumping station into a discrete non-linear knapsack problem and proposed a new hybrid algorithm to improve the computational efficiency of the dynamic programming in solving this problem. Feng et al. [28] studied the optimal operation of parallel pumping stations in an open-channel water transfer system and the influence of the source water level on the discharge. Jing et al. [29] explored the impact of non-electricity costs on the discharge distribution of parallel pumping stations. However, given the enormous difficulties in adjusting the blade angle of the pump, it is difficult to obtain the theoretically optimal discharge. The use of a variable-speed pump (VSP) is more likely to obtain accurate control of the discharge and reduce energy consumption. Marchi et al. [30] introduced a VSP to change the operating point of the pump and found that energy consumption could be significantly reduced. Guo et al. [31] took the minimization of total power consumption as the objective function for the joint operation of a constant-speed pump and VSP and found that the low efficiency of the original pumping station caused by an unreasonable unit combination could be improved.

For some water transfer projects, large pumping stations consisting of blade angle-adjustable pump (BAP) units often operate at low efficiency due to limited water transfer conditions. One measure to improve operation efficiency is to use variable-frequency drives (VFDs) to adjust the speed. However, there is a lack of research on variable speed and blade angle-adjustable pumps (VSBAPs). In this study, a discharge optimization model is proposed for a single pumping station consisting of VSBAP units, and the influence of the number of VFDs is analyzed. Then, a head optimization model is proposed for cascade pumping stations, and its effect on the whole water transfer system is analyzed. As the addition of VFDs will certainly increase the total investment in the water transfer system, the necessity of adding VFDs is also discussed in the Discussion section.

2. Methodology

2.1. Basis of the Model

The head required by a pump to transport water from the inlet port to the outlet port is the sum of the static head of the pump and the total resistance loss of the pipeline, as shown in Equation (1). Since the pump can only operate on the Q-H curve, the stable operating point of the pump is the intersection point of the Q-H curve and the required head curve of the pump, that is, $H=H_r$.

$$H_r=H_{st}+S{Q}^2$$

(1)

where $H_r$ is the required head of the pump (m); $H_{st}$ is the static head of the pump (m), which is the water level difference between upper and lower water surfaces when their pressures are the same; and S is the hydraulic drag coefficient (${\mathrm{s}}^2{/\mathrm{m}}^5$).

It follows from Equation (2) that if the characteristic curve of the VSP can be changed at the rated speed, in other words, if the speed of the BAP is variable, then the characteristic curve corresponding to the angle of each blade of the BAP can be derived to a cluster of characteristic curves at different speeds. Therefore, a BAP equipped with a VFD is applicable to broader operating conditions and can improve the discharge optimization of the pumping station.

$$\frac{Q}{Q_0}=K,\hspace{1em}\frac{H}{H_0}=K^2,\hspace{1em}\frac{P}{P_0}=K^3$$

(2)

where $K=n/n_0$; and $Q_0$, $H_0$, and $P_0$ are the discharge, head, and power of the pump at the rated speed, respectively.

As there are unique Q-H and Q-η curves for a given blade angle, the discharge–head–blade angle and discharge–head–efficiency relationships can be determined. According to the integrated characteristic curve of the pump, the discharge–head–efficiency and discharge–head–blade angle curves are obtained by polynomial fitting, as shown in Equation (3). Substituting Equation (2) into Equation (3) yields the characteristic function under different speeds, as shown in Equation (4). In this case, the efficiency and blade angle are determined only by the discharge, head, and speed ratio. The stable operating head of the pump can be obtained from Equation (1). Considering that the hydraulic loss resulting from the inflow and outflow of the pump in an open-channel water transfer project is negligibly small, it is not considered in this study for the sake of simplicity. Thus, $H=H_r=H_{st}$, where the operating head of the pump is the difference in water levels between the inlet and outlet ports.

$$\eta =F_1\left(Q,H\right),\hspace{1em}\alpha =F_2(Q,H)$$

(3)

$${\eta }^{\prime }=F_1\left(\frac{Q}{K},\frac{H}{K^2}\right),\hspace{1em}{\alpha }^{\prime }=F_2\left(\frac{Q}{K},\frac{H}{K^2}\right)$$

(4)

2.2. Discharge Optimization Model for a Single Pumping Station

(1)

Decision Variables

In this study, the pump equipped with a VFD is referred to as a VUBAP, and the pump not equipped with a VFD is referred to as a BAP for convenience. According to the principle of equal incremental rate, the energy consumption of a pumping station will be minimized under equal discharge distribution if all the pumping units have the same characteristics. It is known from Section 2.1 that the installation of a VFD may change the characteristics of a BAP. Thus, VSBAP and BAP should be regarded as two different types of pumps. In this case, it is natural to assume that the pumping station consisting of both VSBAP and BAP is characterized by the parallel operation of two pumping stations, one consisting of only VSBAP units and the other consisting of only BAP units, and the discharge should be distributed for each type of pumping station independently based on the principle of equal discharge distribution. Therefore, only the total discharge and unit opening number of each pumping station, as well as the speed ratio of the VSBAP, need to be considered in the optimization model.

In summary, the total discharge of VSBAP units, the opening numbers of VSBAP and BAP units, and the speed ratio K of VSBAP units are taken as the decision variables. The total discharge of BAP units is calculated by Equation (5). Figure 1 shows the model structure for one of these cases.

$$Q_{pow}=Q_{t\arg et}-Q_{\mathrm{var}}$$

(5)

where $Q_{t\arg et}$ is the total discharge of the pumping station, and $Q_{pow}$ is the total discharge of BAP units.

(2)

Constraints

Each decision variable should satisfy the constraints described in Equations (6)–(9), where $N_{\mathrm{var},\max }$ and $N_{pow,\max }$ are the total numbers of VSBAP and BAP units; $K_{\min }$ and $K_{\max }$ are the minimum and maximum of the motor speed ratio, respectively.

$$0\le Q_{\mathrm{var}}\le Q_{t\arg et}$$

(6)

$$0\le N_{\mathrm{var}}\le N_{\mathrm{var},\max }$$

(7)

$$0\le N_{pow}\le N_{pow,\max }$$

(8)

$$K_{\min }\le K\le K_{\max }$$

(9)

The discharge and blade angle of each unit should be controlled within a certain range, as shown in Equations (10)–(12), where $q_{pow}$ is the discharge of the BAP; $q_{\mathrm{var}}$ is the discharge of the VSBAP; $q_{\min }$ and $q_{\max }$ are the lower and upper limits of the permissible discharge of the BAP; and ${\alpha }_{\min }$ and ${\alpha }_{\max }$ are the lower and upper limits of the permissible blade angle, respectively.

$$q_{\min }\le q_{pow}\le q_{\max }$$

(10)

$$q_{\min }\cdot K_{\min }\le q_{\mathrm{var}}\le q_{\max }\cdot K_{\max }$$

(11)

$${\alpha }_{\min }\le \alpha \le {\alpha }_{\max }$$

(12)

(3)

Objective function

The goal of the model is to maximize the efficiency of the pumping set while minimizing the overall power consumption of the pumping station. The efficiency of a single pump is calculated as Equation (13):

$${\eta }_{set}={\eta }_{pumb}\cdot {\eta }_{trans}\cdot {\eta }_{motor}\cdot {\eta }_f$$

(13)

where ${\eta }_{set}$ represents the efficiency of the pumping set; ${\eta }_{pumb}$ represents the efficiency of the pump; ${\eta }_{trans}$ represents the efficiency of the gearing connecting the motor and the pump, which is assumed to be 1 when the pump is connected to the motor directly; and ${\eta }_{motor}$ represents the efficiency of the motor, which can be obtained from the manufacturer and is described in Equation (14), where the load rate is defined as the ratio of the motor’s output power to its rated power. As indicated in Equation (15), the shaft power of the motor in this study serves as a proxy for the motor’s output power; q and h stand for the discharge and head of a single pump, respectively; and ${\eta }_f$ is the VFD’s efficiency. It is set to 0.96 in this study since all the values are above the motor’s permitted speed ratio range of 0.96.

$${\eta }_{motor}=F_3\left(\beta \right)$$

(14)

$$\beta =\frac{P_{out}}{P_N}\approx \frac{\rho gqh/{\eta }_{pumb}}{P_N}$$

(15)

The efficiency of the pumping station is the ratio of the output power of all the units to the input power, and the objective function of the optimization model is shown in Equation (16). This paper assumes that the required head is the water level difference between the intake and outlet ports, and the pumping station has a total of n pumping units with VFDs and m pumping units without VFDs. $q_i$ and ${\eta }_{st,i}$ are the discharge and efficiency of the i-th VSBAP unit; and $q_j$ and ${\eta }_{st,j}$ are the discharge and efficiency of the j-th BAP unit, respectively.

$$\max {\eta }_{pumbstation}=\frac{\rho g{Q}_{t\arg et}{H}_r}{{\displaystyle \sum _{i=0}^n\frac{\rho g{q}_i{H}_r}{{\eta }_{st,i}}+{\displaystyle \sum _{j=0}^m\frac{\rho g{q}_j{H}_r}{{\eta }_{st,j}}}}}$$

(16)

(4)

Optimization algorithm

Particle swarm optimization (PSO) is frequently employed to tackle optimization problems because of its strong searching ability, good generality, and fast convergence speed. Although PSO can easily fall into local optimality, it can obtain the approximation of the optimal solution with an allowable error for the discharge optimization model proposed in this paper because the model has a small number of decision variables and these variables are independent of each other and have no strong constraints. The initial population particles within the boundaries of the decision variables in the issue are randomly generated, and the optimal population values are calculated in each iteration to guide the next generation of particles to search in an improved direction until an optimal solution is found.

2.3. Head Optimization Model for Cascade Pumping Stations

Multi-stage low-head pumping stations are connected for long-distance water transfer to a much higher altitude. The efficiency of the entire water transfer project can be profoundly influenced by changes in the water level and discharge of any pumping station due to the intimate hydraulic interactions among these cascade pumping stations. In order to evaluate the impact of the VFD, a head optimization model for cascade pumping stations is proposed in this section. Since hydraulic loss is inevitable for water transport through an open-channel water transfer project [1,32], the hydraulic loss of the channel cannot be neglected in the head optimization of cascade pumping stations. To estimate the hydraulic loss of the channel with precision, Saint-Venant equations are used to simulate the channel head loss under different water level–discharge boundaries [33,34]. The hydraulic loss is described in Equation (17):

$$h_{w,i}=F_4\left(Q_{i+1},Z_{i,in}\right)$$

(17)

where $h_{w,i}$ is the hydraulic loss of the i-th channel; $Q_{i+1}$ is the discharge of the i + 1-th pumping station; and $Z_{i,in}$ is the outlet water level of the i-th pumping station.

As many scholars have studied head optimization of cascade pumping stations, model construction will not be described in this paper. In order to avoid dimensional disaster, the optimal efficiency results of each pumping station under the corresponding operating conditions described in Section 2.2 are used for the objective function. PSO is also used here.

3. Application and Results

3.1. Study Area

The Miyun Reservoir Regulation and Storage Project is an integral component of the South-to-North Water Diversion Project of China, which is aimed at addressing water scarcity in Beijing, as shown in Figure 2. In the Tuancheng Lake–Huairou Reservoir section, water is diverted from Tuancheng Lake to Huairou Reservoir through six-stage pumping stations connected by open channels, and this section is a typical open-channel water transfer system with cascade pumping stations. For the purpose of ensuring the safety of the pumping stations and channels, the discharges of the pump stations at different stages are matched.

In the Tuancheng Lake–Huairou Reservoir section, water is pumped using large-discharge low-head axial or mixed-flow pumps, and the designed water transfer capacity is 20 m³/s. However, as the storage capacities of the channels and the discharges of the pumping stations are adjustable over a small range, there may be mismatches in the discharges for these cascade pumping stations. To be able to address this issue, VFDs are equipped with two pumping units at each pumping station. Under normal operating conditions, three pumps are running simultaneously at each pumping station, and one pump is used as a standby. However, it is difficult to control the discharge of the pump equipped with a VFD, and because of this, it is difficult to ensure an efficient operation of the water diversion project utilizing cascade pumping stations based on personal experience. Therefore, it is significant to obtain the optimal discharge distribution scheme of each pumping station equipped with a VFD in the Tuancheng Lake–Huairou Reservoir section.

Table 1. General information about the pumping stations in the Tuancheng Lake–Huairou Reservoir section.

Pumping Station Name	Design Head (m)	Rated Speed (r/min)	Pump Parameter
			Discharge Range (m3/s)	Head Range (m)
Tundian	1.71	245	5.3–8.2	0.07–1.5
Qianliulin	2.21	245	4.7–7.4	0.67–2.2
Niantou	2.83	245	4.45–6.64	1.05–2.45
Xingshou	2.58	245	4.67–6.86	0.18–2.21
Lishishan	2.21	245	4.9–7.4	0.25–2.04
Xitaishang	6.18	290	4.8–10.3	4.13–8.18

provides the basic parameters of each pumping station. Note that one pumping unit is equipped with only one VFD, and the adjustable speed ratio is in the range of 0.7–1.

3.2. Discharge Optimization for Single Pumping Station

In this section, the Niantou pumping station is analyzed as an example. A series of operating conditions are designed by varying the discharge and head at a step of 0.1 m³/s and 0.01 m, respectively, and the discharge distribution under each condition is calculated based on the discharge optimization model for a single pumping station. The optimization and energy-saving potential of a single pumping station with the addition of VFDs are analyzed, and the influence of the number of VFDs is also discussed.

3.2.1. Discharge Range

Figure 3 shows the optimal operation of pumping units equipped with 0–3 VFDs under different conditions of discharge and head for a single pumping station, where the region corresponding to 0 VSBAPs or BAPs is the non-operable condition. It is clearly observed that the proportion of operable conditions increases as the number of VFDs increases.

Table 2. Operating conditions corresponding to different VFDs.

Number of VFDs	Discharge Range	Average Proportion of Operable Conditions	Relative Increment
0	4.45–19.92 m3/s	42.80%	-
1	3.115–19.92 m3/s	80.14%	87.24%
2	3.115–19.92 m3/s	98.35%	129.8%
3	3.115–19.92 m3/s	98.46%	130%

shows that the operable conditions account for only 42.80% of the total conditions when no VFD is used, while the proportion of operable conditions nearly doubles when one VFD is used and reaches up to 98.35% when two VFDs are used. Nevertheless, no further increase is observed when three VFDs are used.

Because the discharge ranges of 1, 2, and 3 pumping units are not completely overlapped, there is a large non-operable area in Figure 3a. As the pumping unit equipped with a VFD can operate over a larger range of discharge, one VSBAP unit in combination with one BAP unit can be used for the non-operable area corresponding to the discharge range of 8.4–10 m³/s, and for the same reason, one VSBAP unit in combination with two BAP units can be used for the non-operable area corresponding to the discharge range of 13.5~15 m³/s. It is noted that when two VFDs are equipped, almost all the non-operable areas shown in Figure 3a become operable by combining VSBAP and BAP units. Thus, the use of three VFDs would lead to no further increase in the non-operable area. For the BAP unit equipped with a VFD, the discharge range of the unit can be expanded by changing the combination of blade angle and speed, thus increasing the number of operable conditions of the pumping station.

3.2.2. Energy Consumption

Figure 4 shows the efficiency distribution corresponding to the optimal operation of pumping units with different numbers of VFDs, where a darker red indicates a higher efficiency and a darker blue indicates a lower efficiency. As shown in Figure 4a and

Table 3. Optimal efficiency of the pumping station equipped with 0–3 VFDs.

Number of VFDs	Efficiency	Average Efficiency	Average Absolute Increase	Average Relative Increase
0	23.91–63.56%	45.16%	-	-
1	27.25–65.1%	52.25%	8.75%	19.38%
2	32.03–65.1%	57.69%	12.94%	28.65%
3	43.33–65.1%	61.09%	15.86%	35.12%

, the efficiency of the pumping station consisting of only BAP units is generally low, with the average being only 45.16%. As expected, the average efficiency increases with the increase in the number of VFDs under each condition. The absolute and relative increases in the minimum efficiency corresponding to the optimal operation of pumping units are 8.75% and 19.38% on average, while those of the maximum efficiency corresponding to the optimal operation of pumping units are 15.86% and 35.12% on average, respectively. Therefore, equipping the BAP unit with a VFD contributes substantially to increasing the efficiency of the pumping station, and the efficiency increases linearly with the increase in the number of VFDs.

According to Figure 4b–d and

Table 4. Discharge scheme under different operating conditions.

Operating Conditions	Number of VFDs	Discharge (m3/s)			Speed Ratio			Blade Angle (°)			Pumping Station Efficiency (%)
		1#	2#	3#	1#	2#	3#	1#	2#	3#
Discharge = 6 m3/s; head = 1.45 m	0	6			1			−3.46			33.86
	1	6			0.84			2			57.7
	2	6			0.84			2			57.7
	3	6			0.84			2			57.7
Discharge = 10.1 m3/s; head = 2.41 m	0	5.05	5.05		1	1		−3.92	−3.92		50.21
	1	3.46	6.64		0.81	1		−2.71	1.19		60.58
	2	5.05	5.05		0.86	0.86		1.21	1.21		63.35
	3	5.05	5.05		0.86	0.86		1.21	1.21		63.35
Discharge = 16 m3/s, head = 2.05 m	0	5.33	5.33	5.33	1	1	1	−3.93	−3.93	−3.93	44.39
	1	3.11	6.44	6.44	0.74	1	1	−2.9	−0.43	−0.43	54.4
	2	4.68	4.68	6.64	0.8	0.8	1	1.22	1.22	0.2	59.89
	3	5.33	5.33	5.33	0.83	0.83	0.83	2	2	2	65.09

, when one VFD is installed, the efficiency distribution of the pumping station under each condition is basically the same over the discharge range of one unit (3.7–6.5 m³/s). That is, the optimal efficiency can be obtained when a single VSBAP unit is used for water transfer. Over the discharge range of 6.5–11.5 m³/s, the optimal efficiency distribution is basically the same for the pumping station equipped with two and three VFDs. The efficiency of the pumping station is higher under the combination of two VSBAP units compared to that under the combination of one VSBAP unit and one BAP unit. Similarly, over the discharge range of 11.5–20 m³/s, the optimal efficiency is obtained with the use of three VSBAP units.

It is therefore concluded that, under any conditions, when there is a larger number of VSBAP units used in the discharge distribution scheme, the efficiency of the pumping station will be higher and more energy will be saved. It is also found in Table 4 that the VSBAP unit can not only expand the operable conditions of the pumping station, but it can also ensure the high-efficiency operation of the unit under most conditions by changing the combination of the blade angle and speed, which can greatly reduce the energy consumption of the pumping station.

3.3. Head Optimization Model for Cascade Pumping Stations

The influence of the VFD on the water transfer system with cascade pumping stations is further analyzed. A series of operating conditions are designed by varying the discharge at a step of 0.1 m³/s over the range of 10–20 m³/s, and the inlet water level of the first pumping station (Tundian) and the outlet water level of the last pumping station (Xitaishang) are taken as the design values. It is assumed that the number of VFDs is the same across pumping stations. The head distribution scheme under the above conditions is obtained based on the head optimization model.

It is clearly seen in Figure 5 and

Table 5. Number of operable conditions and average efficiency for different numbers of VFDs.

Number of VFDs	Number of Conditions in which the Head Can Be Distributed	Average Efficiency	Absolute Efficiency Gain
0	22	50.37%	-
1	68	55.57%	7.47%
2	85	61.52%	12.04%
3	85	65.84%	15.09%

that, for the original water transfer system with no VFDs, the head can be distributed in the discharge ranges of 12.4–13.2 m³/s and 17.9–20 m³/s, accounting for only 22% of the total conditions, and the efficiency is generally lower than 55%. When one VFD is equipped in each pumping station, the number of operable conditions is increased sharply by 3 times, and the efficiency is in the range of 50–60%; while when two VFDs are equipped in each pumping station, the number of operable conditions is further increased by 3.8 times, accounting for 85% of the total conditions, and the efficiency is generally above 60%. However, further increasing the number of VFDs in each pumping station does not increase the proportion of operable conditions, but the average efficiency is increased to 65.84%.

As each pumping station has the same number of pumping units with similar characteristics, the head distribution for cascade pumping stations is very similar to the discharge distribution for a single pumping station. The operable conditions for head distribution are the intersection of the operable conditions for discharge distribution at each pumping station, and thus the largest head distribution range can be achieved with the use of two VFDs at each pumping station. Increasing the number of VFDs can improve the efficiency of a single pumping station and, consequently, the efficiency of cascade pumping stations. It is also noted that when two or three VFDs are used, the optimal efficiency of a single pumping station is consistent over the discharge range of 10–12 m³/s, and therefore the two curves are overlapped in this discharge range, as shown in Figure 5.

From the results, it is possible to conclude that, from an economic perspective, the most cost-effective strategy is to equip each pumping station with two VFDs. In this case, the operable area is maximized and the efficiency of cascade pumping stations is relatively high, which can achieve the goals of energy saving and emission reduction.

4. Discussion

The installation of VFDs is expected to reduce the energy consumption and eventually the operating cost of a pumping station, but investing in VFDs would cost more in the short run compared to the energy saved. Thus, there is a dispute about the need to install VFDs. In this section, an optimization model is established to determine the number of VFDs at which the sum of the operating cost of the pumping station and the investment in VFDs reaches a minimum. The model is structured as follows: The number of VFDs equipped in each pumping station and the days between the installation of each VFD are selected as decision variables. According to the efficiency of the cascade pumping stations equipped with different numbers of VFDs under different discharges described in Section 3.3 and the electricity price, the electricity cost per day is calculated. Here, the average electricity price is taken as 0.69 CNY/kw∙h for the commercial peak and valley electricity prices in Beijing. The objective function is to minimize the sum of the operating costs of the pumping station and the investment in VFDs. The optimal operation scheme is calculated under different daily discharges within the assumed cost of each group of VFDs (for ease of calculation, the number of VFDs in each group is taken to be 6).

(1)

The influence of VFD cost on the total investment

Figure 6 shows that the total investment for the optimal operation scheme under each discharge increases linearly with the increase in VFD cost, indicating that the total investment can be reduced by installing VFDs at this stage. However, when the cost is greater than the cumulative savings resulting from reduced energy consumption, it is no longer economically viable to install VFDs, and the optimal cost curve becomes horizontal once the threshold is reached. However, it is noted that the threshold differs depending on the daily discharge. As energy consumption is expected to be higher at higher discharges, the turning point of the optimal cost curve would occur earlier at larger discharges.

(2)

The influence of running time on the effect of VFDs

The optimal numbers of VFDs under different running times are calculated, where the average daily discharge is taken to be 18.5 m³/s. As shown in Figure 7, there is no need to install three VFDs within 5 years, irrespective of the cost of the VFDs. As the cost of the VFDs increases, the optimal number of VFDs is changed from two to one, and, in some cases, there may even be no need to install a VFD. As the running time increases, the optimal number of VFDs under different VFD costs is closer to two. That is, the longer the running time, the more energy is saved with the use of two VFDs. To sum up, it is desirable to install two VFDs in the Tuancheng Lake–Huairou Reservoir section for the long-term operation of the cascade pumping stations in an economically profitable way.

5. Conclusions

In order to optimize the operation of BAP units equipped with VFDs, this paper constructs a discharge optimization model and a head optimization model for cascade pumping stations equipped with different numbers of VSBAP units. The results show that the installation of VFDs can increase the number of operable conditions of a single BAP unit by changing the blade angle and speed and ensure the high efficiency of the pumping unit under most operating conditions. With the installation of VFDs, the maximum ratio of the operable conditions in the Tuancheng Lake–Huairou Reservoir section is increased to 3.8 times, and the maximum efficiency is increased by 15.09%. According to the assumed VFD cost, the total investment is significantly reduced with the installation of VFDs. The optimization model constructed in this paper provides an optimal scheme for the operation of single pumping stations and cascade pumping stations equipped with different numbers of VSBAP units and effectively improves the efficiency of large pumping stations consisting of BAP units.