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With pumps as the main devices, the main drainage system (MDS) is critical to mine construction and production. Considering the high cost of traditional manual pump scheduling strategy and the wide adoption of timeofuse (TOU) electricity traffic in coal mines, this paper attempts to reduce the mine operation cost by scheduling the pumps in the flat and valley periods instead of the peak period. For this purpose, the pump scheduling was considered as an optimization problem, the water level of the sump was predicted by double exponential smoothing, and then the optimal pump scheduling plan was derived by ant colony optimization (ACO). The pump scheduling plan obtained by the proposed method was proved cost efficient through experiments on a gold mine in China.
pump scheduling, mine drainage system (MDS), ant colony optimization (ACO), cost efficiency
The mine drainage system (MDS) prevents groundwater and surface water from leaking into the mine during mine construction and production, providing an important guarantee of mine safety against water damage [1]. Most MDSs are controlled manually, i.e. turned on or off by an operator based on his/her experience. The manual control mode consumes lots of electricity. In Chinese coal mining enterprises, the MDSs account for 40 % of the power consumption by all electromechanical devices in coal mines [23]. Since the pump is the centerpiece of each MDS, it is important to design a pump scheduling algorithm to save energy and reduce the relevant cost.
Many pump scheduling methods are available to water distribution systems (WDSs). However, these approaches cannot be applied directly to the MDSs, owing to the following differences between the WDSs and MDSs [4]: the water flow is stable in the WDSs but constantly changing in MDSs; the urban water demand has basically the same daily variation [5], while the mine water inflow mutates from time to time.
To solve the problem, this paper proposes a costeffective pump scheduling (CEPS) algorithm based on water inflow prediction and ant colony optimization (ACO). The main contributions of this paper incudes are as follows: treating the pump scheduling in the MDSs as a discrete optimization problem; developing a water inflow prediction method based on double exponential smoothing to guide the pump scheduling; creating an ACObased algorithm to obtain the most costeffective pump schedule.
The remainder of this paper is organized as follows: Section 2 reviews the previous studies on pump scheduling; Section 3 formulates the pump scheduling problem; Section 4 details the CEPS algorithm; Section 5 applies the CEPS into an actual case of pump scheduling and analyzes the results; Section 6 wraps up this paper with some conclusions.
Pump scheduling is an emerging hotspot in the research of the WDSs, whose water supply relies heavily on pump stations. The pump scheduling of the WDSs is generally treated as an optimization problem, and solved by different optimization algorithms to obtain the (sub)optimal solution. For example, Reference [5] reduces the energy and maintenance costs of WDS pump scheduling by two metaheuristics, simulated annealing (SA) and hybrid genetic algorithm (HGA), and experimentally proves that the SA outperforms the HGA. Reference [6] considers the joint problem of pump scheduling and water flow control as a mixedinteger secondorder cone program, and solves the program with the alternating direction method of multiplier. Reference [7] creates a mixedinteger linear programming model for the scheduling of variablespeed pumps in hydropower stations. Reference [8] models the scheduling of multiple waterlifting pumps in China’s SouthtoNorth Water Diversion Project, which aims to solve the water shortage in northern China, as an optimal operation problem, and solves the problem through dynamic programming. Reference [9] puts forward a similar solution to Mahasawat water distribution station in Thailand. All these energysaving strategies provide good references to the pump scheduling of the MDSs. However, special algorithms should be designed for the MDSs, owing to the said differences between the WDSs and MDSs.
Some of the representative studies on pump scheduling of the MDSs are reviewed as follows. Reference [10] presents a variablespeed hybrid Petri net model of the MDS, and creates on online pump control algorithm based on the model, but the model is difficult to establish due to the MDS variation from mine to mine. Targeting the coal seam 14# in China’s Linnancang Coalmine, Reference [11] sets up a comprehensive model of the seepage field to determine the proper water level in adjacent aquifers, and optimizes the main drainage capacity using the finiteelement subsurface flow system. To improve pump efficiency in the MDSs, Reference [12] establishes a model based on the HGA, but fails to consider the change law of water level. Reference [13] constructs a multipump MDSs optimization model, and applies the artificial bee colony algorithm to determine the number of running pumps in different periods. Reference [14] develops a gray correlation model between water inflow and time, as well as an economic MDS model to implement the load shifting schedule. Based on the water inflow of China’s Fuxin Coalmine, Reference [15] provides a dynamic gray model, and designs an automatic, energyefficient EDS control system. To sum up, the above models all consider MDSs pump scheduling as a continuous optimization problem, which may lead to fragmentation of pump running time. The frequent start and stop of pumps will exacerbate equipment aging. Therefore, this paper views the pump scheduling as a discrete optimization problem, aiming to strike a balance between the number of pump starts/stops and the electricity consumption.
Table 1 lists the main symbols and their definitions in this paper. A typical MDS consists of a water sump to store the mineral water, and several pumps to drain the water from the sump when the water level surpasses the preset threshold. Because the electricity consumption varies from period to period in one day, the power supply usually adopts the timeofuse (TOU) electricity tariff mechanism to minimize the pressure on the grid. As shown in Figure 1, the TOU electricity tariff divides one day (24 hours) into several periods, and the electricity tariff in each period may falls into the valley, flat or peak segment. In this case, the pump scheduling is to determine the pump running period that minimizes the electricity consumption and control the water level in the sump under the preset threshold.
Table 1. Symbols and definitions
Symbol 
Definition 
H_{t} 
Water level at time t 
$\phi$ 
The predefined threshold of water level 
K 
Number of pumps of an MDS 
p 
Power of a pump 
L 
Length of predefined time period (unit: minute) 
N 
Number of time periods in 24 hours, L·N=1440 minutes 
C 
Cost of pumps in 1 day 
c_{p},_{ }c_{f}, c_{v} 
Electricity tariff in peak, flat, and valley segment, respectively 
n_{s} 
Number of running pumps in sth period, 0≤s≤N 
c_{s} 
Electricity tariff in sth period 
H_{s} 
Water level at the beginning of sth period 
F_{s} 
Increased water level by water inflow in sth period 
D_{s} 
Decreased water level by draining water with pumps in sth period 
F_{s}^{(1)} 
Basic exponential smoothing value of Fs 
F_{s}^{(2)} 
Double exponential smoothing value of Fs 
$\widehat{H}_{s}$ 
Predicted value of H_{s} 
d 
Decreased water level by one pump in one time period 
$\omega$ 
Smoothing factor to compute F_{s}^{(1)} and F_{s}^{(2)} 
G=(V,E,W) 
Multistage directed and weighted graph for pump schedule 
$v_{j}^{(s)} \in \mathrm{V}$ 
A vertex of G, which is in sth stage 
$v_{s}^{(0)}, v_{e}^{(N+1)} \in \mathrm{V}$ 
Additional first and last vertex of G 
V_{s} 
Set of vertices of sth stage 
w_{i,j} 
Weight of edge $\left\langle v_{i}^{(s)}, v_{j}^{(s+1)}\right\rangle$ 
M 
Number of ants of ACO 
$\tau_{i, j}$ 
Pheromone laying on edge $\left\langle v_{i}^{(s)}, v_{j}^{(s+1)}\right\rangle$ 
$\eta_{i, j}$ 
Locally available heuristic information of edge $\left\langle v_{i}^{(s)}, v_{j}^{(s+1)}\right\rangle$ 
$\rho_{i, j}^{A}$ 
Probability of ant A at $v_{i}^{(s)}$ to choose $v_{j}^{(s+1)}$ to visit 
$C_{b e s t}$ 
Cost of the best pump schedule 
$\alpha, \beta, \rho$ 
Parameters used in ACO 
ItrMax 
Maximum number of iterations of ACO 
Figure 1. The TOU electricity tariff
Let H_{t} be the water level of the sump at time t, ∅ be the preset threshold of water level, and K be the number of pumps of an MDS. Meanwhile, it is assumed that the pumps have the same, nonadjustable power p, each day (24 hours) encompasses N periods of equal length L and the same electricity tariff, and the electricity tariffs in peak, flat, and valley segments are c_{p}, c_{f} and c_{v}, respectively. Then, the daily pump cost of the MDS can be expressed as:
$\mathcal{C}=\sum_{i=1}^{K} \int_{0}^{24}(p \cdot r(t) \cdot c(t)) d t$ (1)
where r(t)=0 if the pump is off at time t, and r(t)=1 if the pump is on at time t; c(t) is the electricity tariff at time t. Since each day is divided into several periods, equation (1) can be transformed into:
$\mathcal{C}=\sum_{s=1}^{N} p \cdot n_{s} \cdot c_{s} \cdot L$ (2)
where n_{s} and c_{s} are the number of running pumps and electricity tariff in period s, respectively. Therefore, the pump scheduling problem can be defined as:
$\min \mathcal{C}=\sum_{s=1}^{N} p \cdot n_{s} \cdot c_{s} \cdot L \cdot s \cdot t$
$0 \leq n_{s} \leq K, c_{s} \in\left\{c_{p}, c_{f}, c_{v}\right\}, 0 \leq H_{t} \leq \phi$ (3)
If the water level of the sump is predictable, then the problem defined in equation (3) is to find the n_{s} at the beginning of each period, such that the solution space contains (K+1)^{N} feasible solutions. This task is hard to solve by brute force. This paper adopts the ACO to complete the task. The ACO provides a desirable way to solve discrete optimization problems [15]. This algorithm is inspired by the behavior of real ant colonies: when an ant colony wants to find the shortest path between their nest and a food source, the ants constantly release pheromones, directing each other to resources, while exploring their environment. Each ant constructs a feasible solution and updates the pheromones according to the quality of solution, and the pheromones guide the ants to construct a better solution in the next loop.
4.1 Water level prediction
As mentioned before, each day can be divided into several periods of equal length; in each period, each pump is either in the on state or the off state. Hence, it is necessary to determine the water level at the beginning of each period. Let H_{s} be the water level at the beginning of period s. Then, the water level at the subsequent period can be calculated as:
H_{s+1}=H_{s}+F_{s}D_{s} (4)
where F_{s} is the water level increase induced by the water inflow; D_{s} is the water level decrease induced by the water drainage. The value of D_{s} is already known, as the pump parameters are given in advance. The pumps should be turned on if H_{s+1} is greater than the preset threshold on water level $\phi$. Since the F_{s} is constantly changing, the double exponential smoothing was introduced to predict its value.
Let $\omega$ be the smoothing factor. Then, the basic exponential smoothing value of the F_{s} can be described as:
$\mathrm{F}_{\mathrm{s}+1}^{(1)}=\omega \mathrm{F}_{\mathrm{s}}+(1\omega) \mathrm{F}_{\mathrm{s}}^{(1)}$ (5)
The double exponential smoothing value of the F_{s} can be described as:
$\mathrm{F}_{\mathrm{s}+1}^{(2)}=\omega \mathrm{F}_{\mathrm{s}+1}^{(1)}+(1\omega) \mathrm{F}_{\mathrm{s}}^{(2)}$ (6)
Then, the predicted value of F_{s+j} can be obtained as:
$\widehat{F}_{s+j}=a_{s}+b_{s} j$ (7)
where
$\left\{\begin{array}{c}{a_{s}=2 F_{s}^{(1)}F_{s}^{(2)}} \\ {b_{s}=\frac{\omega}{1\omega}\left(F_{s}^{(1)}F_{s}^{(2)}\right)}\end{array}\right.$ (8)
The value of D_{s} depends on the number of running pumps n_{s} and the water level reduction d caused by one pump in each period:
$D_{s}=n_{s} d$ (9)
Therefore, the predicted value of $\widehat{H}_{S+1}$ can be written as:
$\widehat{\mathrm{H}}_{\mathrm{s}+1}=\mathrm{H}_{\mathrm{s}}+\hat{\mathrm{F}}_{\mathrm{s}}\mathrm{n}_{\mathrm{s}} \mathrm{d}$ (10)
4.2 ACObased pump scheduling
Before solving the pump scheduling problem by the ACO, the problem defined in equation (3) must be transformed into the shortest path problem in a graph.
As shown in Figure 2, the pump scheduling problem can be transformed into a multistage directed and weighted graph $\mathrm{G}=(\mathrm{V}, \mathrm{E}, \mathrm{W})$, where $\mathrm{V}=\left\{v_{j}^{(s)}  s=1,2, \cdots, N ; j=0,1, \cdots, K .\right\} \cup\left\{v_{s}^{(0)}, v_{e}^{(N+1)}\right\}$, $\mathrm{E}=\left\{\left\langle v_{i}^{(s)}, v_{j}^{(s+1)}\right\rangle\right\} \cup\left\{\left\langle v_{s}^{(0)}, v_{i}^{(1)}\right\rangle\right\} \cup\left\{\left\langle v_{i}^{(N)}, v_{e}^{(N+1)}\right\rangle\right\}$ and $\mathrm{W} : \mathrm{E} \rightarrow \mathbb{R}$ is the weight of edges ( $\mathbb{R}$ is the set of real numbers).
Each period corresponds to one stage in G, and each stage has K vertices corresponding to the K pumps. Let $V_{s}(i=1,2, \cdots, N)$ be the set of vertices in stage s. Then, $\mathrm{V}_{s}=\left\{v_{i}^{(s)}  i=0,1, \cdots, K\right\}$. The weight of $\left\langle v_{i}^{(s)}, v_{j}^{(s+1)}\right\rangle$, w_{i,j}, is the cost incurred by turning on j pumps and turning off i pumps, with $w_{s, i}=w_{i, e}=0(i=1,2, \cdots, K)$. Therefore, the pump scheduling problem is equivalent to finding the shortest path from $v_{s}^{(0)}$ to $v_{e}^{(N+1)}$, which can be solved by the ACO.
Figure 2. The graph of the pump scheduling problem
The basic procedure of the ACObased pump scheduling plan is as follows:
Step 1: Solution construction. Initially, M ants are all placed at $v_{s}^{(0)}$. In each iteration, each ant chooses the next vertex at a certain probability. For ant A at vertex $v_{i}^{(s)}$, the probability of the ant to visit $v_{j}^{(s+1)}$ can be expressed as:
$\rho_{\mathrm{i}, \mathrm{j}}^{\mathrm{A}}=\frac{\tau_{\mathrm{i}, \mathrm{j}}^{\alpha} \eta_{\mathrm{i}, \mathrm{j}}^{\beta}}{\sum_{\mathrm{j}=0}^{\mathrm{K}} \tau_{\mathrm{i}, \mathrm{j}}^{\alpha} \eta_{\mathrm{i}, \mathrm{j}}^{\beta}}$ (11)
where $\tau_{i, j}$ and $\eta_{i, j}=\frac{1}{w_{i, j}}$ are the pheromone and local heuristic information of $\left\langle v_{i}^{(s)}, v_{j}^{(s+1)}\right\rangle$, respectively; $\alpha$ and $\beta$ are the relative importance parameters of $\tau_{i, j}$ and $\eta_{i, j}$, respectively. If several vertices have the same probability, the ant will select one of them by random.
In addition, the solution construction of each ant must satisfy the constraints in equation (3). The first constraint, $0 \leq n_{s} \leq K$, is automatically satisfied, because $0 \leq\left\mathrm{V}_{s}\right \leq K$; the second constraint, $c_{s} \in\left\{c_{p}, c_{f}, c_{v}\right\}$, is used to compute the cost of each solution; the third constraint, $0 \leq H_{t} \leq \phi$, is satisfied if the next vertex is not $v_{0}^{(s+1)}$ if ant A is at vertex $v_{i}^{(s)}$, and $\widehat{H}_{s+1}>\phi$ or $\widehat{H}_{s+2}>\phi$.
Step 3: Pheromone update. After all ants have constructed their solutions, the pheromone trails are updated by the following rule:
$\tau_{i, j}=(1\rho) \tau_{i, j}+\Delta \tau_{i, j}^{b e s t}$ (12)
where $0<\rho \leq 1$ is the pheromone evaporation rate; $\Delta \tau_{i, j}^{b e s t}$ is the amount of pheromone released by the ants on $\left\langle v_{i}^{(s)}, v_{j}^{(s+1)}\right\rangle$, which can be defined as
$\Delta \tau_{i, j}^{b e s t}=\left\{\begin{array}{cc}{\frac{1}{C_{b e s t}}} & {\text { if }\left\langle v_{i}^{(s)}, v_{j}^{(s+1)}\right\rangle \text { is in the best solution }} \\ { 0} & {\text { otherwise }}\end{array}\right.$ (13)
Based on the above description, the ACObased pump scheduling algorithm was summed up as follows:
Algorithm 1. CEPS algorithm

Function CEPS (G)// G is the graph as Figure 2 

Set parameters and initialize pheromone trails; 

for Itr $\leftarrow 1$ to ItrMax 

for $i \leftarrow 1$ to M 

List $_{0}^{\mathrm{i}} \leftarrow \mathrm{v}_{\mathrm{s}}^{(0)}$ 

for $j \leftarrow 1$ to N+1 

$\mathbf{u} \leftarrow$ List $_{j1}^{\mathrm{i}}$ 

$\mathrm{v} \leftarrow \operatorname{argmax}\left\{\rho_{\mathrm{u}, \mathrm{v}}^{\mathrm{i}}  \rho_{\mathrm{u}, \mathrm{v}}^{\mathrm{i}} \text { computed by }(11) \text { and v satisfying constraints of $(3)$}\right\}$ 

List $_{\mathrm{j}}^{\mathrm{i}} \leftarrow \mathrm{v}$ 

end for 

end for 

List$_{\text { best }}^{\text { Itr }}$ $\leftarrow$ shortest path of this iteration 

$C_{\mathrm{best}}^{\mathrm{Itr}} \leftarrow \operatorname{cost}$ of List $_{\mathrm{best}}^{\mathrm{Itr}}$ 

List$_{\text { best }}^{\text { global }}$ $\leftarrow$ shortest path so far 

$\mathcal{C}_{\text { best }}^{\text { global }} \leftarrow$ cost of List$_{\text { best }}^{\text { global }}$ 

update $\tau$ according to (12) and (13) 

end for 

return List$_{\text { best }}^{\text { global }}$ and $\mathcal{C}_{\text { best }}^{\text { global }}$ 

end function 
From September 1^{st} to 30^{th}, 2018, several experiments were carried out on the real data of Jinchiling Gold mine in Zhaoyuan, eastern China’s Shandong Province, to verify the feasibility of the proposed algorithm. One MDS with K=5 pumps was selected for the experiments from the mine. The power of each pump is p=110 kW. The water level threshold of the sump is $\phi=2.2 \mathrm{m}$. The TOU electricity tariff is given in Table 2, where c_{p} , c_{f} and c_{v} are respectively RMB 1.252yuan, 0.782 yuan and 0.370 yuan. Each day (24h) was divided evenly into N=72 periods with the length L=20 minutes. The smoothing factor was set to $\omega=0.7$. The symbols and their definitions were given in Table 3 below.
Table 2. TOU electricity tariff
Time period 
0:006:00 
6:008:00 
8:0011:00 
11:0018:00 
18:0021:00 
21:0024:00 
Tariff (Unit: Yuan) 
0.370 
0.782 
1.252 
0.782 
1.252 
0.370 
Table 3. Symbols and definitions
Period 
1 
2 
3 
4 
5 
6 
7 
8 
9 
F_{s} 
2.095 
2.107 
2.118 
2.126 
2.135 
2.147 
2.158 
2.169 
2.180 
$\hat{F}_{s+1}$ 
2.095 
2.112 
2.126 
2.135 
2.144 
2.158 
2.169 
2.180 

$\hat{F}_{s+2}$ 
2.095 
2.118 
2.135 
2.143 
2.153 
2.168 
2.180 

$\hat{F}_{s+3}$ 
2.095 
2.124 
2.144 
2.152 
2.162 
2.179 

Period 
10 
11 
12 
13 
14 
15 
16 
17 
18 
F_{s} 
2.188 
2.200 
2.209 
2.220 
2.221 
2.221 
2.302 
2.310 
2.322 
$\hat{F}_{s+1}$ 
2.190 
2.197 
2.210 
2.220 
2.230 
2.228 
2.225 
2.336 
2.341 
$\hat{F}_{s+2}$ 
2.191 
2.201 
2.207 
2.221 
2.230 
2.241 
2.234 
2.227 
2.376 
$\hat{F}_{s+3}$ 
2.191 
2.201 
2.212 
2.216 
2.232 
2.240 
2.251 
2.240 
2.230 
Period 
19 
20 
21 
22 
23 
24 
25 
26 
27 
F_{s} 
2.329 
2.337 
2.348 
2.356 
2.364 
2.373 
2.382 
2.391 
2.403 
$\hat{F}_{s+1}$ 
2.343 
2.343 
2.347 
2.358 
2.365 
2.373 
2.382 
2.390 
2.400 
$\hat{F}_{s+2}$ 
2.369 
2.362 
2.355 
2.357 
2.368 
2.374 
2.381 
2.391 
2.399 
$\hat{F}_{s+3}$ 
2.417 
2.397 
2.381 
2.367 
2.366 
2.378 
2.383 
2.390 
2.400 
Figure 3. Relative errors of predicted values
Figure 3 records the relative errors of the predicted values of 27 water inflow levels (F_{s}) in one day. It can be seen that $\hat{F}_{s+1}$ is the most accurate. The mean relative errors of $\hat{F}_{s+1}$, $\hat{F}_{s+2}$ and $\hat{F}_{s+3}$ are, respectively, 0.039 %, 0.066 %, and 0.091 %. The water level variation at 16 causes the greatest prediction error.
Using the above prediction data, the pump scheduling was carried out by Algorithm 1. For convenience, the original pump scheduling plan is denoted as the original plan, and the pump scheduling plan after the optimization by Algorithm 1 is denoted as the optimized plan. Note that the latter plan is simulated rather than the real scheduling of Jinchiling gold mine.
The ACO parameters were directly extracted from Reference [16], which also finds the shortest path in a graph by the ACO, including M = 30 ants, $\alpha=1$, $\beta=2$, $\rho=0.9$ and $\operatorname{ItrMax}=300$. The costs of original and optimized plans are compared in Table 4 and Figure 4. The comparison shows that Algorithm 1 can reduce the mine operation cost by 34.09 % on average from the level of the original plan.
Table 4. Total costs of the original and optimized plans (Unit: RMB yuan)
Day 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Original 
7507 
6562 
7474 
7931 
6678 
8611 
7461 
7553 
9084 
8641 
Optimized 
4230 
3671 
4696 
5547 
5079 
4781 
5165 
5369 
6389 
6559 
Day 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
Original 
7932 
6254 
8785 
6823 
6307 
7835 
6502 
7533 
4655 
7198 
Optimized 
5274 
4702 
5458 
5413 
4127 
6061 
3456 
3850 
3560 
4015 
Day 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
Original 
7888 
8190 
6426 
6246 
7608 
7423 
6813 
7721 
6304 
7010 
Optimized 
5151 
4661 
3480 
4019 
4562 
5666 
4274 
5362 
4504 
4981 
Figure 4. Cost comparison between the original and optimized plans
(In each pair of bars, the left and right bars are respectively the costs of the original and optimized plans.)
Table 5 shows the machine hours of the pumps in peak, flat and valley periods of the two plans. It can be seen that the optimized plan greatly reduces the machinehours in peak period, and increases the machinehours in valley period. The results are consistent with Figure 4, where the cost in peak period only accounts for a fraction of the total cost.
Table 5. Machinehours of the pumps in the original and optimized plans in different periods
Time period 
Peak 
Flat 
Valley 
Original 
17.67 
39.21 
54.62 
Optimized 
2.65 
27.99 
98.55 
With pumps as the main devices, the MDSs often consume lots of electricity in mine production. In most mines, the TOU electricity tariff mechanism is adopted because the pumps only start when the water level in the sump reaches a preset threshold. Thus, pump scheduling is a possible way to optimize the MDS energy efficiency. This paper utilizes double exponential smoothing method to predict the water inflow, and employs the ACO to obtain the optimal pump scheduling plan. The proposed pump scheduling method was verified through experiments in a gold mine in China. The experimental results show that the optimized plan can greatly reduce the mine operation cost. The future research will explore the pump scheduling of MDSs with different types of pumps.
This paper is made possible thanks to the National Key Research and Development Plan of China (Grant No.: 2018YFC1406203; 2017YFC0804406)
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