Real-time gate pump control method for regulating and controlling urban flood

Abstract

The invention discloses a gate pump real-time control method for regulating and controlling urban flood. The method comprises the following steps: S1, collecting urban river data; s2, constructing an urban hydrological model, and simulating an urban rainfall-runoff forming process according to detailed space division and different surface runoff characteristics; s3, constructing a river network hydrodynamic model, and simulating a river network system by considering hydraulic conditions of pump stations, gates, weir flows and reservoir and lake facilities; s4, the hydrological model describing the urban surface runoff and confluence process and the hydrodynamic model describing the water flow evolution process in the urban river network are coupled through the section node water exchange process to construct a hydrological and hydrodynamic coupling model; s5, compiling a water engineering real-time control rule based on the warning water level and ecological water level data of the river channel; s6, inputting the real-time control rule of the water engineering into the hydrological and hydrodynamic coupling model, evaluating the performance of the method through a rainstorm event, and quantitatively analyzing the response of flood to the real-time control of the water engineering; according to the invention, accurate regulation and control of the river network and effective relief of flood are realized.

Description

Technical Field

[0001] The present invention relates to the technical field of water resource management and flood control, and in particular to a real-time control method for a sluice pump for controlling urban floods. Background Art

[0002] Sluice gates and pumps are key facilities for regulating water regimes in urban river networks, undertaking important tasks such as drainage scheduling and water resource management. Currently, with climate change and intensified urbanization, urban hydrological processes are becoming more complex, with significantly increased uncertainty in upstream water inflow and rainfall events. However, in their actual operation, traditional sluice gate and pump control models, due to their over-reliance on manual experience-based decision-making, control rules lagging behind water regime changes, and insufficient coordination between sluice gates and pumps, have failed to fully develop and utilize these systems, making them difficult to adapt to changing flood control and drainage needs and potentially exacerbating urban flooding risks. Therefore, developing a set of autonomous, scientific, and systematic sluice gate and pump control rules is key to improving the regulatory capabilities and operational efficiency of urban water engineering facilities. Summary of the Invention

[0003] The purpose of the present invention is to overcome the above-mentioned shortcomings and provide a real-time control method for gate pumps to regulate urban flooding, so as to regulate the water level of the river network during heavy rain and thus alleviate the risk of flooding.

[0004] To solve the above technical problems, the technical solution adopted by the present invention is: a real-time control method for a sluice pump for regulating urban flooding, comprising the following steps:

[0005] S1, collect urban river data;

[0006] S2, construct an urban hydrological model to simulate the urban rainfall-runoff formation process based on detailed spatial division and different surface runoff characteristics;

[0007] S3, construct a river network hydrodynamic model, considering the hydraulic conditions of pumping stations, gates, weirs, reservoirs and lakes, and simulate the river network system;

[0008] S4, the hydrological model describing the urban surface runoff generation and confluence process is coupled with the hydrodynamic model describing the flow evolution process in the urban river network through the process of water volume exchange at cross-section nodes to construct a hydrological and hydrodynamic coupling model;

[0009] S5, compile real-time control rules for water projects based on the river's warning water level and ecological water level data;

[0010] S6, input the real-time control rules of water projects into the hydrological and hydrodynamic coupling model, evaluate the performance of the method based on rainstorm events, and quantitatively analyze the response of floods to the real-time control of water projects.

[0011] Preferably, in S1, the collected river network data include: river network topology relationship, cross-section data, underlying surface, gate and pump station data required to construct the river network system, the gate and pump station data include design parameters and distribution locations, measured rainfall data, measured river network water level and flow data corresponding to the measured rainfall data.

[0012] Preferably, the hydrological and hydrodynamic coupling model in S4 is constructed based on SWMM, and the specific construction process is as follows:

[0013] S41, call the hydrology module in SWMM to calculate surface runoff. For single short-duration rainfall events, rainwater evaporation is not considered.

[0014] S42, call the hydrology module in SWMM to calculate the land surface infiltration process, using Horton infiltration;

[0015] In S43, call the hydrology module in SWMM and use the nonlinear reservoir method to calculate the confluence. The Manning equation and the continuity equation are solved to obtain the following equation:

[0016]

[0017] Where Δt is the time step; d1 is the initial water depth; d2 is the final water depth; d p is the filling amount; n is the Manning roughness coefficient; i * is the net rainfall; L is the sub-basin overflow width; A1 is the surface area;

[0018] S44, calling the hydrodynamic module in SWMM, calculates the river network confluence by completely solving the Saint-Venant equations using dynamic waves;

[0019] S45 uses the gate model provided by SWMM to calculate gate discharge, including the calculation method of submerged outflow and non-submerged outflow;

[0020] S46, uses SWMM to provide a constant speed pump model, and the flow rate is the design flow rate of the pump station;

[0021] S47, the surface runoff and river network flow are coupled by exchanging water at cross-section nodes.

[0022] Preferably, in S41, the calculation formula for runoff generation of different underlying surfaces is as follows:

[0023] Permeable areas:

[0024] R1=(if)TD;

[0025] Where R1 is the runoff of the permeable area; i is the rainfall intensity; f is the rainwater infiltration rate; D is the maximum depression storage capacity; T is the rainfall duration;

[0026] Impervious areas with depression storage:

[0027] R2 = PD;

[0028] Where R2 is the runoff of the impervious area with depression storage; P is the accumulated rainfall; D is the maximum depression storage;

[0029] For impermeable areas without depression storage, all but a small amount of rainfall evaporates and is converted into runoff.

[0030] Preferably, in S42, the Horton osmosis is performed as follows:

[0031]

[0032] where f p is the infiltration capacity; t is the time from the beginning of rainfall to the current moment; f ∞ is the minimum or stable infiltration capacity, i.e., the infiltration capacity at t = ∞; f0 is the maximum or initial infiltration capacity, i.e., the infiltration capacity at t = 0; k d is the attenuation coefficient.

[0033] Preferably, in S44, the river network confluence is calculated by completely solving the Saint-Venant equations using dynamic waves, as shown in the following formula:

[0034] Continuity equation:

[0035]

[0036] Momentum equation:

[0037]

[0038] Where x is the distance; t is the time; A2 is the flow cross-sectional area; Q is the flow rate of the river network cross section; H is the head of the river channel; S f is the friction drop; g is the acceleration due to gravity.

[0039] Preferably, the calculation method of the submerged outflow and the non-submerged outflow in S45 is as follows:

[0040] Flooded outflow:

[0041]

[0042] Non-submerged outflow:

[0043]

[0044] where Q g is the gate overflow flow; C d is the discharge coefficient; A0 is the area of ​​the gate opening; g is the acceleration of gravity; H eis the effective water head of the gate; ω is the gate opening; Y full is the gate fully open height; H1 is the water head in front of the gate; Z0 is the gate low elevation.

[0045] Preferably, in S47, the water exchange relationship of a certain river network section is as follows:

[0046]

[0047] in is the flow rate of the river network section at time t; is the flow rate of the river network section at time t-1; is the surface runoff corresponding to the river network section at time t.

[0048] Preferably, the basis for writing the real-time control rules for the water project in S5 is the warning water level and ecological water level data of the river:

[0049]

[0050] in, is the upper limit water level regulated by the i-th gate; is the lower limit water level regulated by the i-th gate; is the upper limit water level controlled by the jth pump station; is the lower limit water level controlled by the jth pump station; is the warning water level and ecological water level of the control reference section of the i-th gate; is the warning water level of the control reference section of the j-th pump station; Δh j is the control threshold of the control reference section of the j-th pump station.

[0051] Preferably, the real-time control rule of the water project input into the hydrological and hydrodynamic coupling model in S6 is expressed as:

[0052] S61, gate control rules are:

[0053]

[0054] THEN GATE N=0

[0055]

[0056] THEN GATE N=1

[0057] Among them, NODE N is the gate control reference section ID; 0 means the gate is closed, and 1 means the gate is fully open;

[0058] S62, pump station control rules are:

[0059]

[0060] THEN PUMP M STATUS=OFF

[0061]

[0062] THEN PUMP M STATUS=ON

[0063] Among them, NODE M is the control reference section ID of the pump station; OFF means the pump station is closed, and ON means the pump station is open;

[0064] S63, using rainstorm events to evaluate the performance of the method, quantitatively analyzing the response of floods to real-time control of water projects, includes the rainwater well water level process and the number of overflows, the river section water level process and the number of overflows.

[0065] Beneficial effects of the invention: In view of the current situation that water engineering facilities in existing urban rivers are not fully utilized for dynamic regulation, the invention formulates joint regulation rules for gates and pumps based on data such as the warning water level and ecological water level of the rivers in the target area, and simulates and calculates the operation scenarios of water engineering facilities through hydrological and hydrodynamic models. It can regulate the water level of the river network during heavy rains and thus alleviate the risk of floods, thereby achieving precise regulation of the river network and effective relief of floods; the invention improves the efficiency and accuracy of existing urban gate and pump regulation, takes into account the needs of the ecological environment while ensuring flood control safety, and has good real-time and applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 This is a schematic diagram of the study area in Cangshan District, Fuzhou City;

[0067] Figure 2 Comparison of simulated and observed water level processes for stormwater wells and river channel cross-section samples;

[0068] Figure 3 The heavy rain and tidal process of Typhoon Soudelor;

[0069] Figure 4 The overflow of rainwater wells and river sections is not regulated for the project;

[0070] Figure 5 Regulate the water level process of rainwater wells and river section samples for water projects;

[0071] Figure 6 To regulate the overflow of rainwater wells and river sections for water projects. DETAILED DESCRIPTION

[0072] The present invention will be described in further detail below with reference to the accompanying drawings and specific embodiments.

[0073] Example 1:

[0074] This embodiment involves a real-time control method for a sluice pump for regulating urban flooding, comprising the following steps:

[0075] S1. Collect river network data, including river network topology, cross-section data, underlying surface, gate and pump station data required for constructing the river network system. Gate and pump station data include design parameters and distribution locations, measured rainfall data, and measured river network water level and flow data corresponding to the measured rainfall data.

[0076] S2, construct an urban hydrological model to simulate the urban rainfall-runoff formation process based on detailed spatial division and different surface runoff characteristics;

[0077] S3, construct a one-dimensional hydrodynamic model of the river channel, considering the hydraulic conditions of facilities such as pumping stations, gates, weirs, reservoirs and lakes, and simulate the river network system;

[0078] S4, the hydrological model describing the urban surface runoff generation and confluence process is coupled with the hydrodynamic model describing the flow evolution process in the urban river network through the process of water volume exchange at cross-section nodes to construct a hydrological and hydrodynamic coupling model;

[0079] S5, compile real-time control rules for water projects based on the river's warning water level and ecological water level data;

[0080] S6, input the real-time control rules of water projects into the hydrological and hydrodynamic coupling model, evaluate the performance of the method based on rainstorm events, and quantitatively analyze the response of floods to the real-time control of water projects.

[0081] In this embodiment, the urban hydrological model in S2, the river network hydrodynamic model in S3, and the hydrological-hydrodynamic coupling model in S4 are constructed based on SWMM. SWMM is an existing model. The hydrological-hydrodynamic coupling in this embodiment is constructed by calling the hydrological model and hydrodynamic model of SWMM. The specific construction process is as follows. It should be noted that since the specific calculation and solution process of the relevant modules belongs to the existing technology, this embodiment only lists the model construction principle and does not list the detailed solution process.

[0082] S41, call the hydrology module in SWMM to calculate surface runoff. For a single short-duration rainfall event, rainwater evaporation is not considered. The runoff calculation formula for different underlying surfaces is as follows:

[0083] Permeable areas:

[0084] R1=(if)TD

[0085] Where R1 is the runoff of the permeable area; i is the rainfall intensity; f is the rainwater infiltration rate; D is the maximum depression storage capacity; T is the rainfall duration;

[0086] Impervious areas with depression storage:

[0087] R2=PD

[0088] Where R2 is the runoff of the impervious area with depression storage; P is the accumulated rainfall; D is the maximum depression storage;

[0089] For impervious areas without depression storage, except for a small amount of rainfall evaporation, the rest is all converted into runoff;

[0090] S42, call the hydrology module in SWMM to calculate the land surface infiltration process, using Horton infiltration, as follows:

[0091]

[0092] where f p is the infiltration capacity; t is the time from the beginning of rainfall to the current moment; f ∞ is the minimum or stable infiltration capacity, i.e., the infiltration capacity at t = ∞; f0 is the maximum or initial infiltration capacity, i.e., the infiltration capacity at t = 0; k d is the attenuation coefficient;

[0093] In S43, call the hydrology module in SWMM and use the nonlinear reservoir method to calculate the confluence. The Manning equation and the continuity equation are solved to obtain the following equation:

[0094]

[0095] Where Δt is the time step; d1 is the initial water depth; d2 is the final water depth; d p is the filling amount; n is the Manning roughness coefficient; i * is the net rainfall; L is the sub-basin overflow width; A1 is the surface area;

[0096] S44, call the hydrodynamic module in SWMM and calculate the river network confluence by using dynamic waves to completely solve the Saint-Venant equations, as shown below:

[0097] Continuity equation:

[0098]

[0099] Momentum equation:

[0100]

[0101] Where x is the distance; t is the time; A2 is the flow cross-sectional area; Q is the flow rate of the river network cross section; H is the head of the river channel; S f is the friction drop; g is the acceleration due to gravity;

[0102] S45, using the gate model provided by SWMM to calculate the gate discharge, including the calculation method of submerged outflow or non-submerged outflow, where:

[0103] Flooded outflow:

[0104]

[0105] Non-submerged outflow:

[0106]

[0107] where Q g is the gate overflow flow; C d is the discharge coefficient; A0 is the area of ​​the gate opening; g is the acceleration of gravity; H e is the effective water head of the gate; ω is the gate opening; Y full is the gate fully open height; H1 is the water head in front of the gate; Z0 is the gate lower elevation;

[0108] S46, uses SWMM to provide a constant speed pump model, and the flow rate is the design flow rate of the pump station;

[0109] S47, using the cross-section node water exchange method to couple surface runoff and river network flow, the water exchange relationship of a certain river network section is as follows:

[0110]

[0111] in is the flow rate of the river network section at time t; is the flow rate of the river network section at time t-1; is the surface runoff corresponding to the river network section at time t.

[0112] In this embodiment, the basis for writing the real-time control rules for water projects in S5 is the warning water level and ecological water level data of the river:

[0113]

[0114] in, is the upper limit water level regulated by the i-th gate; is the lower limit water level regulated by the i-th gate; is the upper limit water level controlled by the jth pump station; is the lower limit water level controlled by the jth pump station; is the warning water level and ecological water level of the control reference section of the i-th gate; is the warning water level of the control reference section of the j-th pump station; Δh j is the control threshold of the control reference section of the j-th pump station.

[0115] Correspondingly, the real-time control rule of the water project input into the hydrological and hydrodynamic coupling model in S6 is expressed as:

[0116] S61, gate control rules are:

[0117]

[0118] THEN GATE N=0

[0119]

[0120] THEN GATE N=1

[0121] Among them, NODE N is the gate control reference section ID; 0 means the gate is closed, and 1 means the gate is fully open;

[0122] S62, pump station control rules are:

[0123]

[0124] THEN PUMP M STATUS=OFF

[0125]

[0126] THEN PUMP M STATUS=ON

[0127] Among them, NODE M is the control reference section ID of the pump station; OFF means the pump station is closed, and ON means the pump station is open;

[0128] S63, using rainstorm events to evaluate the performance of the method, quantitatively analyzing the response of floods to real-time control of water projects, includes the rainwater well water level process and the number of overflows, the river section water level process and the number of overflows.

[0129] Example 2:

[0130] The method of the present invention is further described below by taking a certain area of ​​Nantai Island, Fuzhou City, Fujian Province as an example. The area of ​​the area is 39.17 km 2 The existing geographical, spatial, and geometric information of more than 100 river sections and 1,000 river sections in the region, as well as the location information and design parameters of four pumping stations and nine gates, are provided. The parameter settings of the hydrological and hydrodynamic model constructed based on SWMM are shown in Table 1.

[0131] Table 1 Model parameter value range

[0132]

[0133] This example uses two rainstorm events from 14:00 on March 26, 2022 to 4:00 on March 27, 2022 and from 0:00 to 12:00 on April 27, 2022 to calibrate and verify the urban river network hydrological and hydrodynamic coupling model, respectively. These events are recorded as Event 1 and Event 2, with rainfall amounts of 68.3 mm and 126.0 mm, respectively. Three rainwater wells (M1, M2, M3) and three river sections (S1, S2, S3) were randomly selected from the observation points as samples. The average NSE of the sample heads under Event 1 and Event 2 were higher than 0.85 and 0.79, respectively. The model simulation effect is good and can be used for river network regulation calculation and analysis. The comparison of the simulated head and observed head process of the rainwater well samples and river section samples is shown in Figure 2 .

[0134] This example considers the scenario without water engineering control (referred to as Scenario A) and uses the urban river network hydrological and hydrodynamic coupling model to simulate the "Soudelor" typhoon rainstorm event ( Figure 3 ) flood process. Figure 4 As shown, the inundation depth of a rainwater well is the difference between the well water level and the ground elevation at which the well is located. The inundation depth of a river section is the difference between the river section water level and the minimum elevation of the left and right banks at that section. The number of overflows from rainwater wells and river sections was 1,545 and 323, respectively, with overflow rates of 30.26% and 30.13%, respectively. In the absence of controlled urban water projects, heavy rain is the primary factor affecting flooding. Therefore, real-time control of water projects in areas severely affected by flooding is crucial.

[0135] The real-time control rules for water projects are compiled based on the river warning water level, ecological water level, and regulation threshold data provided by relevant units. See Table 2:

[0136] Table 2 Gate and pump control rules

[0137]

[0138] Furthermore, this example considers the water engineering control scenario (Scenario B) and uses the urban river network hydrological and hydrodynamic coupling model to simulate the flood process of the "Soudelor" typhoon rainstorm event. Figure 5 As shown in the figure, compared with scenario A, the average water level process value of the samples under scenario B decreased by 35.05%, the peak water level decreased by 9.02%, and the number of peak water level times decreased significantly. Figure 6 As shown in the figure, compared with scenario A, the overflow of the drainage system around the river network in scenario B is improved. In scenario B, the number of overflows of rainwater wells and river sections are 1796 and 639 respectively, and the overflow proportions are 35.18% and 59.61% respectively.

[0139] From this, it can be seen that real-time control of water projects during heavy rains played an important role in flood control. The overflow of drainage systems around the river network was significantly reduced, alleviating flood disasters.

[0140] The above embodiments are merely preferred technical solutions of the present invention and should not be construed as limiting the present invention. The scope of protection of the present invention shall be the technical solutions set forth in the claims, including equivalent alternatives to the technical features of the technical solutions set forth in the claims. In other words, equivalent alternatives and improvements within this scope are also within the scope of protection of the present invention.

Claims

1. A real-time control method for a sluice pump for regulating urban flooding, characterized by: The following steps are involved: S1, collect urban river data; S2, construct an urban hydrological model to simulate the urban rainfall-runoff formation process based on detailed spatial division and different surface runoff characteristics; S3, construct a river network hydrodynamic model, considering the hydraulic conditions of pumping stations, gates, weirs, reservoirs and lakes, and simulate the river network system; S4, the hydrological model describing the urban surface runoff generation and confluence process is coupled with the hydrodynamic model describing the flow evolution process in the urban river network through the process of water volume exchange at cross-section nodes to construct a hydrological and hydrodynamic coupling model; S5, compile real-time control rules for water projects based on the river's warning water level and ecological water level data; S6, input the real-time control rules of water projects into the hydrological and hydrodynamic coupling model, evaluate the performance of the method based on rainstorm events, and quantitatively analyze the response of floods to the real-time control of water projects.

2. A real-time control method for a sluice pump for regulating urban flooding according to claim 1, characterized in that: In S1, the collected river network data include: river network topology required for constructing the river network system, cross-section data, underlying surface, gate and pump station data, gate and pump station data including design parameters and distribution locations, measured rainfall data, measured river network water level and flow data corresponding to the measured rainfall data.

3. The method for real-time control of a sluice pump for regulating urban flooding according to claim 1, characterized in that: The hydrological and hydrodynamic coupling model in S4 is constructed based on SWMM. The specific construction process is as follows: S41, call the hydrology module in SWMM to calculate surface runoff. For single short-duration rainfall events, rainwater evaporation is not considered. S42, call the hydrology module in SWMM to calculate the land surface infiltration process, using Horton infiltration; In S43, call the hydrology module in SWMM and use the nonlinear reservoir method to calculate the confluence. The Manning equation and the continuity equation are solved to obtain the following equation: Where Δt is the time step; d1 is the initial water depth; d2 is the final water depth; d p is the filling amount; n is the Manning roughness coefficient; i * is the net rainfall; L is the sub-basin overflow width; A1 is the surface area; S44, calling the hydrodynamic module in SWMM, calculates the river network confluence by completely solving the Saint-Venant equations using dynamic waves; S45 uses the gate model provided by SWMM to calculate gate discharge, including the calculation method of submerged outflow and non-submerged outflow; S46, uses SWMM to provide a constant speed pump model, and the flow rate is the design flow rate of the pump station; S47, the surface runoff and river network flow are coupled by exchanging water at cross-section nodes.

4. The method for real-time control of a sluice pump for regulating urban flooding according to claim 3, characterized in that: In S41, the calculation formula for runoff generation on different underlying surfaces is as follows: Permeable areas: R1=(if)TD; Where R1 is the runoff of the permeable area; i is the rainfall intensity; f is the rainwater infiltration rate; D is the maximum depression storage capacity; T is the rainfall duration; Impervious areas with depression storage: R2 = PD; Where R2 is the runoff of the impervious area with depression storage; P is the accumulated rainfall; D is the maximum depression storage; For impermeable areas without depression storage, all but a small amount of rainfall evaporates and is converted into runoff.

5. The method for real-time control of a sluice pump for regulating urban flooding according to claim 3, characterized in that: In the S42, Horton's infiltration is performed as follows: where f p is the infiltration capacity; t is the time from the beginning of rainfall to the current moment; f ∞ is the minimum or stable infiltration capacity, i.e., the infiltration capacity at t = ∞; f0 is the maximum or initial infiltration capacity, i.e., the infiltration capacity at t = 0; k d is the attenuation coefficient.

6. The method for real-time control of a sluice pump for regulating urban flooding according to claim 3, characterized in that: In the above S44, the river network confluence is calculated by using dynamic waves to completely solve the Saint-Venant equations, as shown in the following formula: Continuity equation: Momentum equation: Where x is the distance; t is the time; A2 is the flow cross-sectional area; Q is the flow rate of the river network cross section; H is the head of the river channel; S f is the friction drop; g is the acceleration due to gravity.

7. The real-time control method for a sluice pump for regulating urban flooding according to claim 3, characterized in that: The calculation method of the submerged outflow and non-submerged outflow in S45 is as follows: Flooded outflow: Non-submerged outflow: where Q g is the gate overflow flow; C d is the discharge coefficient; A0 is the area of ​​the gate opening; g is the acceleration of gravity; H e is the effective water head of the gate; ω is the gate opening; Y full is the gate fully open height; H1 is the water head in front of the gate; Z0 is the gate low elevation.

8. The real-time control method for a sluice pump for regulating urban flooding according to claim 3, characterized in that: In S47, the water exchange relationship of a certain river network section is as follows: in is the flow rate of the river network section at time t; is the flow rate of the river network section at time t-1; is the surface runoff corresponding to the river network section at time t.

9. The method for real-time control of a sluice pump for regulating urban flooding according to claim 1, characterized in that: The basis for writing the real-time control rules of the water project in S5 is the warning water level and ecological water level data of the river: in, is the upper limit water level regulated by the i-th gate; is the lower limit water level regulated by the i-th gate; is the upper limit water level controlled by the jth pump station; is the lower limit water level controlled by the jth pump station; is the warning water level and ecological water level of the control reference section of the i-th gate; is the warning water level of the control reference section of the j-th pump station; Δh j is the control threshold of the control reference section of the j-th pump station.

10. The real-time control method for gate pumps for regulating urban flooding according to claim 1, characterized in that: The real-time control rule of the water project input into the hydrological and hydrodynamic coupling model in S6 is expressed as: S61, gate control rules are: THEN GATE N=0 THEN GATE N=1 Among them, NODE N is the gate control reference section ID; 0 means the gate is closed, and 1 means the gate is fully open; S62, pump station control rules are: THEN PUMP M STATUS=OFF THEN PUMP M STATUS=ON Among them, NODE M is the control reference section ID of the pump station; OFF means the pump station is closed, and ON means the pump station is open; S63, using rainstorm events to evaluate the performance of the method, quantitatively analyzing the response of floods to real-time control of water projects, includes the rainwater well water level process and the number of overflows, the river section water level process and the number of overflows.

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