A Cost-effective Pump Scheduling Method for Mine Drainage System Based on Ant Colony Optimization

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 meta-heuristics, 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 mixed-integer second-order cone program, and solves the program with the alternating direction method of multiplier. Reference [7] creates a mixed-integer linear programming model for the scheduling of variable-speed pumps in hydropower stations. Reference [8] models the scheduling of multiple water-lifting pumps in China’s South-to-North 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 energy-saving 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 variable-speed 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 finite-element 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 multi-pump 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, energy-efficient 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.

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 pre-set 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 ACO-based 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 multi-stage 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.

2.png

Figure 2. The graph of the pump scheduling problem

The basic procedure of the ACO-based 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 ACO-based pump scheduling algorithm was summed up as follows:

Algorithm 1. CEPS algorithm

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 pre-set 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.