Keyword

1 Introduction

With the development of Chinese economy and society, the electricity load is increasing gradually. It is particularly important to ensure the safe and stable operation of regional substations and improve the ability of disaster prevention and reduction of substations to ensure the power supply demand of our country [1]. At present, the calculation and evaluation of pipe network flood control and drainage mainly include hydrology [2] and hydrodynamics [3]. Among them, hydrology method has a simple calculation process and easy access to required data. However, its description of the physical process of water flow is insufficient, and it is difficult to accurately describe the spatial-temporal variation characteristics of water flow in pipe network system [4]. By solving the Saint-Venant equations (dynamic wave) and its simplified equations (diffusion wave and moving wave), the hydrodynamic method can calculate the flow process in the pipe network [5]. This kind of method can accurately simulate the physical process of water flow with high calculation accuracy, and it has been widely used in the calculation of pipe network water flow [6,7,8]. Open full flow (i.e. the coexistence or alternate flow of unpressurized flow and pressurized flow) is a water flow phenomenon that often occurs in the process of pipeline drainage, and it is one of the hot spots and difficulties in the analysis and simulation of pipe network flow [9]. However, at present, the analysis and calculation of the open full flow process of the pipe network are mainly focused on the diversion tunnel, the pressure discharge of the reservoir and the urban flood control, and there are few studies on the substation waterlogging prevention and control. Therefore, based on the basic equation of open full flow (continuity) and momentum, combined with Preissmann slit hypothesis, the one-dimensional open full flow governing equation and discrete form are derived in this paper. On this basis, the flow control test and simulation of groundwater drainage pipe in substation are carried out. The research results will provide scientific basis for substation flood disaster loss assessment, environmental risk control, flood control and other planning analysis and formulation.

2 One-Dimensional Bright Full Flow Model and Discretization

2.1 Mathematical Model

The continuity and momentum equation of unsteady flow in one-dimensional open channel is [10, 11]:

$$ \frac{\partial Z}{{\partial t}} + \frac{1}{B}\frac{\partial Q}{{\partial x}} = 0 $$
(1)
$$ \frac{\partial Q}{{\partial t}} + \frac{\partial }{\partial x}(Qv) + gA\left( {\frac{\partial Z}{{\partial x}} + S_f } \right) = 0 $$
(2)

The continuity and momentum equation of one-dimensional pressurized unsteady flow is:

$$ \frac{\partial H}{{\partial t}} + \frac{c^2 }{{gA}}\frac{\partial Q}{{\partial x}} = 0 $$
(3)
$$ \frac{\partial (Q)}{{\partial t}} + \frac{\partial }{\partial x}(Qv) + gA\left( {\frac{\partial H}{{\partial x}} + S_f } \right) = 0 $$
(4)

where, Z is the water level, H is the pipe flow head, B is the section width, Q is the flow, Sf is the friction slope, which can be calculated by Manning formula, c is the wave velocity, v is the velocity.

In 1961, Preissmann proposed that the narrow slit method is an effective method for the calculation of pressurized and nonpressurized interfaces. It is assumed that there is an extremely narrow gap at the top of the pipe, which does not increase the area and hydraulic radius of the pipe. When the water depth is higher than the top of the tube, the area of wet circumference and water through section remains unchanged. The narrow slit method is used to treat the pressurized pipe flow equivalent, and the slit width is set as B (Fig. 1), then the slit width can be calculated by the formula \(B = \frac{gA}{{c^2 }}\), Where, A is the total area of the section. The water level in the control equation of open channel unsteady flow is represented by the water head H of pressure tube in the control equation of pressure unsteady flow, and \(\frac{c^2 }{{gA}} = \frac{1}{B}\), then the governing equation of one-dimensional open channel unsteady flow and one-dimensional pressure unsteady flow can be written in the following form:

$$ \frac{\partial Z}{{\partial t}} + \frac{1}{B}\frac{\partial Q}{{\partial x}} = 0 $$
(5)
$$ \frac{\partial Q}{{\partial t}} + \frac{\partial }{\partial x}(Qv) + gA\left( {\frac{\partial Z}{{\partial x}} + S_f } \right) = 0 $$
(6)

Considering local head loss and lateral inflow, the governing equation of one-dimensional bright full flow model can be obtained according to Eqs. (5) and (6) as follows:

$$ \frac{\partial A}{{\partial t}} + \frac{\partial Q}{{\partial x}} = q_0 $$
(7)
$$ \frac{\partial Q}{{\partial t}} + \frac{\partial }{\partial x}(Qv) + gA\left( {\frac{\partial Z}{{\partial x}} + S_f + S_L } \right) = 0 $$
(8)
Fig. 1.
figure 1

Preissmann slit schematic diagram

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2.2 One-Dimensional Bright Full Flow Equation Discretization

By means of staggered grid and semi-implicit discretization, the motion equation is discretized on the river segment (pipe segment), and the continuity equation is discretized on the node. This discrete scheme can not only ensure the conservation and stability of the format, but also facilitate the calculation of data input. Figure 2 shows the spatial staggered grid discrete diagram, in which I represents nodes and i represents units. It is assumed that each section of river network or pipe network has (N + 1) nodes and N units.

Fig. 2.
figure 2

Equation discretization diagram

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Dispersion of Momentum Equation.

Equation (8) is integrated on the unit i, and the obtained formula is semi-implicit discrete, which can be obtained as follows:

$$ \begin{array}{*{20}l} {(Q_i^{t + \Delta t} - Q_i^t )\frac{\Delta x_i }{{\Delta t}} + v_{I + 1} Q_{I + 1}^{t + \theta \Delta t} - v_I Q_I^{t + \theta \Delta t} + gA_i (S_{fi} + S_{Li} )\Delta x_i } \hfill \\ { = gA_i (Z_I^{t + \theta \Delta t} - Z_{I + 1}^{t + \theta \Delta t} )} \hfill \\ \end{array} $$
(9)

where, \(\Delta x_i\) is the length of the unit i, \(\theta\) is the implicit coefficient, the format has unconditional stability when \(\theta\) > 5, \(S_{Li}\) is the local head loss, which can be deduced from \(S_{Li} = \frac{|Q_i |Q_i }{{gC_{cs} A_{cs} \Delta x_i }}\), where \(C_{cs}\) is the flow coefficient of the building, \(A_{cs}\) is the water flow area of the building, \(S_{fi}\) is the head loss along the road.

In order to increase the stability of the discrete scheme, a first-order upwind interpolation scheme is used for node flow.

$$ v_{I + 1} Q_{I + 1}^{t + \theta \Delta t} = \left\{ \begin{gathered} v_{I + 1} Q_i^{t + \theta \Delta t} \ \ \ \ \ \ \ \ \ \ \ \ \ v_{I + 1} > 0 \hfill \\ v_{I + 1} Q_{i + 1}^{t + \theta \Delta t} \ \ \ \ \ \ \ \ \ \ \ \ {\text{Other}} \hfill \\ \end{gathered} \right. $$
(10)
$$ v_I Q_I^{t + \theta \Delta t} = \left\{ \begin{gathered} v_I Q_{i - 1}^{t + \theta \Delta t} \ \ \ \ \ \ \ \ \ \ \ \ \ v_I > 0 \hfill \\ v_I Q_i^{t + \theta \Delta t} \ \ \ \ \ \ \ \ \ \ \ \ {\text{Other}} \hfill \\ \end{gathered} \right. $$
(11)

Substituting the above two formulas into Eq. (9), we can get:

$$ \begin{gathered} a_i Q_{i - 1}^{t + \Delta t} + b_i Q_i^{t + \Delta t} + c_i^k Q_{i + 1}^{t + \Delta t} \hfill \\ = P_i + \theta gA_i (Z_I^{t + \Delta t} - Z_{I + 1}^{t + \Delta t} ) \hfill \\ \end{gathered} $$
(12)

Among them:

\(a_i = - \theta \max (v_I ,0)\),

$$ b_i = \frac{\Delta x_i }{{\Delta t}} + \theta \left[ {\max (v_{I + 1} ,0) + \max ( - v_I ,0)} \right] + \frac{{gn_i^2 \left| {Q_I } \right|\Delta x_i }}{{A_I R_I^{{\raise0.7ex\hbox{$4$} \!\mathord{\left/ {\vphantom {4 3}}\right.\kern-0pt}\!\lower0.7ex\hbox{$3$}}} }} $$

\(c_i = - \theta \max ( - v_{I + 1} ,0)\),

$$ \begin{gathered} P_i = - (1 - \theta )\left[ {\max (v_{I + 1} ,0) + \max ( - v_I ,0)} \right]Q_i + (1 - \theta )\max ( - v_{I + 1} ,0)Q_{i + 1} \hfill \\ + (1 - \theta )\max ( - v_I ,0)Q_{i - 1} + (1 - \theta )gA_i (Z{\,}_{Ik} - Z_{I + 1} ) + Q_i \frac{\Delta x_i }{{\Delta t}} \hfill \\ \end{gathered} $$

Discretization of Continuity Equation.

Integrate Eq. (7) on node I, as follows:

$$ \Delta x_i B_i \frac{{Z_I^{t + \Delta t} - Z_I^t }}{\Delta t} + Q_{I - 1}^{t + \theta \Delta t} = (q_{mh} )_i $$
(13)

where, \(\Delta x_I B_I = \frac{{\Delta x_{i - 1} B_{i - 1} }}{2} + \frac{\Delta x_i B_i }{2} + A_{mZ,i}\), \(q_{mh}\) indicates the node traffic, \(A_{mZ,i}\) indicates the node area. Substituted into Eq. (13), we can get:

$$ Q_i^{t + \Delta t} - Q_{i - 1}^{t + \Delta t} + E_I Z_I^{t + \Delta t} = D_I $$
(14)

Among them:

$$ E_{Ik} = \frac{1}{\theta \Delta t}\left( {\frac{{\Delta x_{i - 1} B_{i - 1} }}{2} + \frac{\Delta x_i B_i }{2} + A_{mh,i} } \right) $$
$$ D_{Ik} = \frac{1}{\theta }\left[ { - \frac{{Z_{Ik} }}{\Delta t}\left( {\frac{{\Delta x_{i - 1} B_{i - 1} }}{2} + \frac{\Delta x_i B_i }{2} + A_{mh,i} } \right) - (1 - \theta )Q_i^t + (1 - \theta )Q_{i - 1}^t } \right] $$

3 Test and Simulation of Water Flow Control of Underground Drainage Pipe in Substation

In this paper, a 35 kV substation located in Qiandongnan Prefecture, Guizhou Province is selected to carry out the water flow control experiment of underground drainage pipeline. As shown in Fig. 3, the underground water pipe of the substation is L-shaped, with the length of the transverse and longitudinal pipes being 12.2 m and 13.5 m respectively. The pipe is buried in the central position of the transformer station's electric field with a depth of 1 m. Pipe shape and hydraulic parameters are shown in Table 1. During the control test, water was injected from the transverse pipe and outflow was observed in the standpipe (Fig. 3). The water injection was controlled by an automatic pumping device, and the water injection time was 500s. The water injection per unit time presented a linear change, as shown in Fig. 4. An automatic flow monitor is used at the outlet of the standpipe to observe the outflow. The outflow process is shown in Fig. 5.

Fig. 3.
figure 3

Substation underground drainage pipeline

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Fig. 4.
figure 4

Control the inflow process of the drainage pipeline in the test

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Fig. 5.
figure 5

Process simulation of control test drainage pipeline outflow

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Table 1. Shape and hydraulic parameters of underground drainage pipelines
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The flow process of pipeline was simulated by using the deduced one-dimensional model and discrete model, and the simulation results were evaluated by Nash efficiency coefficient (NSE). The results show that the model can well calculate the pipeline flow migration process (Fig. 5), and the NSE is 0.9. The observed peak discharge is 3.6 L/s, and the simulated peak discharge is 3.7 L/s, indicating that the model can accurately capture the flood discharge and provide technical support for the analysis and early warning of flood control and drainage in the substation.

4 Conclusion

Flood control and drainage of substation are of great significance to ensure the demand of power supply. In this paper, drainage of underground drainage pipeline of substation is taken as the research object, and the actual substation pipeline is used to carry out the water flow control test. On this basis, the hydrodynamics method is used to carry out the simulation calculation, and the accuracy of the calculation of pipeline drainage is analyzed and evaluated. Based on the basic equation of open full flow (continuity) and momentum, combined with Preissmann slit hypothesis, the one-dimensional open full flow governing equation and its discrete form are derived, and the one-dimensional open full flow calculation model is established. The model is verified by the test results of water flow control in underground drainage pipe of substation. The results show that the model can simulate the pipeline flow process and flood peak flow well, and provide technical support for flood control and drainage early warning of substation in wet area.