A Review of Primary Mine Ventilation System Optimization

Within the mining industry, a safe and economical mine ventilation system is an essential component of all underground mines. In recent years, research scientists and engineers have explored operations research methods to assist in the design and safe operation of primary mine ventilation systems. The main objective of these studies is to develop algorithms to identify the primary mine ventilation systems that minimize the fan power costs, including their working performance. The principal task is to identify the number, location, and duty of fans and regulators for installation within a defined ventilation network to distribute the required fresh airflow at minimum cost. The successful implementation of these methods may produce a computational design tool to aid mine planning and ventilation engineers. This paper presents a review of the results of a series of recent research studies that have explored the use of mathematical methods to determine the optimum design of primary mine ventilation systems relative to fan power costs. This paper provides (1) a definition of the objectives that govern the design and operation of a main mine ventilation system, (2) a summary of the interconnected tunnels that form the mine network, and (3) a review of the engineering components (including fans and regulators) that distribute the fresh air within the mine network. We present a mathematical formulation of the objective function and constraints that define the optimization problem, and follow this with a review of the optimization methods proposed to find the best engineering solution. The optimization problem seeks to minimize the ventilation power consumption of a defined network, subject to the conservation of mass flow and energy, the satisfaction of upper and lower airflow bounds, and any controls on the location of fans and regulators (Barnes 1989). Large mine networks can be complex; they often comprise hundreds of interconnected roadways, which are ventilated by fans and regulators positioned around the network. Minimizing the power costs of these systems is a challenging, practical problem that may be examined as a solution to a highly nonlinear mathematical problem.

rules specify the minimum safety and environmental conditions that a ventilation system must maintain within a working underground mine. The main objectives of such a system are to sustain life and to ensure a safe and acceptable working environment. Achieving these aims requires the delivery of a sufficient quantity of fresh air to ensure the rapid dilution of mine pollutants to below statutory occupational exposure levels (OELs). These pollutants may include strata gases (e.g., methane and radon), mineral dust and fumes, and particulates from dieselpowered equipment.
Mine ventilation engineers currently use personal experience and good practice guidelines to identify the best mine ventilation arrangements by the repeated use of steady state mine ventilation network programs to evaluate the airflow and pressure distribution. However, this method is time consuming and may not always identify the solution that delivers the minimum fan power cost. The challenge for the engineer is to identify a practical subset of ventilation arrangements and performance levels that deliver the desired airflow distribution. The use of any new ventilation arrangement needs to satisfy all mining, health, and safety laws of the country in which the mine operates. For example, in the United States, any significant changes to a mine ventilation system need the prior approval of the Mine Safety and Health Administration.
National mining laws can differ between countries. For example, the European Union allows booster fans in coal mines, whereas the United States prohibits them.
The principal optimization problem associated with underground mine ventilation systems is to determine the number, location, and duty of the fans and regulators to deliver the needed airflow and pressure distribution at the lowest fan power or energy consumption.
This paper provides (1) a definition of the objectives that govern the design and operation of a main mine ventilation system, (2) a summary of the interconnected tunnels that form the mine network, and (3) a review of the engineering components (including fans and regulators) that distribute the fresh air within the mine network. We present a mathematical formulation of the objective function and constraints that define the optimization problem, and follow this with a review of the optimization methods proposed to find the best engineering solution. The optimization problem seeks to minimize the ventilation power consumption of a defined network, subject to the conservation of mass flow and energy, the satisfaction of upper and lower airflow bounds, and any controls on the location of fans and regulators (Barnes 1989). Large mine networks can be complex; they often comprise hundreds of interconnected roadways, which are ventilated by fans and regulators positioned around the network. Minimizing the power costs of these systems is a challenging, practical problem that may be examined as a solution to a highly nonlinear mathematical problem.

Primary Mine Ventilation Systems
The layout of interconnected tunnels formed to access and exploit the underground mineral deposits defines the topology of an underground mine ventilation network. The design of a ventilation system is determined principally by the location and rates of generation of the contaminants within the mine circuit. Most pollutant emissions occur at the working faces. The working face is that portion of a tunnel in which minerals are actively extracted. Although the design of each individual mine ventilation system is unique, each system has similar infrastructural and operational characteristics. Vertical shafts or inclined tunnels, commonly termed ramps, connect the underground network of tunnels to the surface. These surface connections serve as (1) egress routes for workers, mobile machinery, and materials, (2) conveyance paths for the extracted mineral, and (34) entry or exit portals for the ventilation air. Figure 1 shows a generic layout of a mine ventilation network, including typical locations of doors and airflow distribution controls represented by regulators and fans.
Separating the flow paths provided by the tunnels that form a mine network into a fresh (intake) and a contaminated (return) flow circuit is normal. Roadways fitted with airlock doors interconnect these two flow circuits. The double D (DD) symbol in Figure 1 represents airlock doors. These doors prevent the short circuit of the fresh air to the contaminated return air circuit, and allow for the safe and efficient movement of workers, equipment, and supplies between the two airflow circuits. To create a controlled flow of fresh air to all working areas, engineers supplement any natural thermal airflow drafts with large main fans, normally located at or near the surface. These main fans either push fresh air into, or pull contaminated air out of, the underground mine ventilation network. Booster fans and air regulator doors divide the fresh airflow entering the mine between the various mine workings. The B and R symbols in Figure 1 represent the locations of a booster fan and airflow regulators, respectively. The regulators throttle the airflow passing through a particular airway and distribute the fresh airflow delivered by the main fan.
Booster fans are added when the main fan(s) cannot deliver the pressure to transport the needed airflow quantities to the furthest locations in the mine. This may occur when a mine ventilation network expands to exploit mineral reserves at an increased distance or depth from the surface connections. Where local mining laws and regulations allow, underground booster fans may be installed within selected branches to deliver the energy to overcome the pressure losses within the circuit and preserve the required ventilation flow rates. The correct design and installation of booster fans should minimize uncontrolled recirculation, but not adversely influence the performance of other fans installed in the network.
Within a mine, there are working locations in which the pollutant necessitates a minimum airflow quantity to dilute it below statutory OELs. These working locations are termed specified airflow branches. The mining activities that produce pollutants include mineral production faces, diesel maintenance workshops, battery and fuel charging stations, and mineral crusher stations.

Mathematical Representation
A directed graph composed of nodes and edges, termed junctions and branches by ventilation practitioners (Moll and Lowndes 1992), mathematically represents the topography of a primary mine ventilation network (PMVN). The edges represent the different paths airflow can follow and the nodes are the connections between those paths. Mine ventilation network calculations employ the defined network topography, apply Kirchhoff's first and second circuit laws, and ensure that the pressure drops and airflows in each branch satisfy the Atkinson equation. These laws ensure that the calculated airflow distribution complies with the principles of mass and energy conservation (McPherson 1993, Lowndes andTuck 1995). Equations 2 to 15 in the appendix mathematically model these physical principles.
Kirchhoff's first law states the mass flow entering a junction (node) is equal to the mass flow leaving that junction. Equation (15)  Kirchhoff's second law states that the sum of the directed energy differences (i.e., potentials and losses) around a closed mesh is zero. In a ventilation network, the energy potentials are the pressure energies delivered by the fans and the thermal drafts. The frictional resistance and the shock losses create the energy losses experienced within the airways or roadway. Changes in the roadway cross section, flow around a bend in the roadway, and flow around equipment within the roadway generate changes to the flow direction or speed. The changes in the flow direction or speed result in energy losses, termed shock losses. The shock pressure losses, caused by changes in the flow direction or speed within the j th branch, are often estimated as the frictional pressure drop created by an equivalent length of the host branch. This equivalent length ( equiv j s ) is then added to the actual physical length of the host airway to calculate the total frictional and shock pressure drop experienced within this branch.
For incompressible and turbulent flow, Equation (3) represents the frictional and shock pressure losses (Lj) experienced within the j th branch, which is proportional to the square of the volumetric flow rate (Qj) in that branch. The constant of proportionality, termed the total resistance of branch (rj), is determined from the evaluation of Atkinson's equation (McPherson 1993), as Equation (4) represents. The total resistance (rj) is equal to the product of the friction factor (kj), the perimeter (pj), and the sum of the length and equivalent shock loss length (sj + equiv j s ) of the branch, divided by the cube of the cross-sectional area (Aj) of the branch.
Equation (5) represents the shock pressure loss (Rj) because of a regulator within the j th branch. We can also estimate Rj as the frictional pressure drop created by an equivalent length ( reg j s ) of the host branch. We determine the resistance generated by the regulator ( * j r ) in the host airway from the evaluation of the Atkinson's equation; see Equation (6).
When heat is transferred to the air, a thermal draft, termed the natural ventilation pressure, produces pressure differences within a mine network. High temperature sources that produce a flow of heat to the air include the surrounding rock and installed equipment. Thus, for an incompressible and turbulent airflow, the total pressure drop (Hj), experienced within the j th branch of a mine network, is determined by the sum of four pressure drop components: the frictional pressure loss (Lj), the shock pressure loss of a regulator (Rj), the fan pressure gain (Fj), and the natural ventilation pressure gain (Nj). Equation (7) represents the total pressure drop in the j th branch. Kirchhoff's second circuit law states that the algebraic sum of all the total pressure drops experienced around the i th closed path in the network (termed the i th fundamental mesh Mi) is equal to zero. Equation (8)   The arrow alongside each branch in Figure 2 shows the direction of airflow. The curved arrow at the center of the mesh indicates the chosen direction of the flow in the mesh. If Nb is the total number of branches and Nn is the total number of nodes in the ventilation network, then Nm (Nm = Nb -Nn +1) defines the number of fundamental meshes (closed paths) present in the network. In Equation (8), bij is the coefficient that multiplies the pressure drop on each branch (Hj); bij = 1 if the i th mesh contains the j th branch and the flow within this branch is in the same direction as that defined for the mesh; bij = − 1 if the i th mesh contains the j th branch and the flow within this branch is in the opposite direction as that defined for the mesh; and bij = 0 if the i th mesh does not contain the j th branch.
For a given mine layout, the engineers determine the location and airflow requirements of each working zone within the mine. The rates of mineral production and transport define the levels of pollutants created in the mine. This knowledge allows the ventilation engineer to determine the minimum and maximum airflow quantities delivered by the ventilation system (McPherson 1993). Equations (9) and (10)  Additionally, Equations (11) and (12) represent the fan air power and energy consumption, respectively. Pj is the delivered air power and j is the total efficiency of the fan located in the j th branch of the network. However, practical constraints may preclude the installation of regulators or fans within certain mine roadways. For example, booster fans are not normally installed within major transport or mineral conveyance roadways. To exclude a regulator or booster fan within the j th branch, the regulator pressure loss term (Rj) or the fan pressure term (Fj) is zero.
For a given set of initial conditions, the linear and nonlinear equations defined by Equations (2) to (8) are solved, using commercially available programs, to evaluate the steady state distribution of airflows and pressures within a given mine ventilation network. The most popular iterative method used to perform this steady state evaluation of a mine ventilation network airflows and pressures is the Hardy Cross method (Cross 1936). The Hardy Cross method is an example of a single-step Newton-Raphson numerical scheme. The algorithm first identifies an initial feasible solution for the balanced flows within the network. The method then iteratively determines the corrections needed to balance the calculated pressure losses around each mesh in the network and calculates the changes needed to the flow estimates. The process repeats until consecutive solutions satisfy a chosen convergence criterion. For specified airflow branches, the method determines the value of the equivalent resistance or fan pressure added to this branch to achieve the required minimum airflow quantity.
Several commercial mine ventilation network programs are available to evaluate the steady state volumetric flow and pressure distribution within a mine ventilation network. The evaluation of the steady state airflow and pressure distribution can be manually calculated using the Hardy Cross method, but it is a tedious and time-consuming process. Table 1 summarizes examples of these programs (Hardcastle 1995  Source: Calizaya et al. (1987).

Primary Mine Ventilation Network Optimization Problem
The PMVN optimization problem is defined by the objective function presented in Equation (1) coupled to the system of equations defined by Equations (2) to (14), detailed in the appendix.
The objective is to minimize the fan power cost of a mine ventilation network, while delivering the airflow distribution required by the mine activities. Equations (13) and (14) represent the nonnegativity and nature of the variables, respectively. The PMVN optimization problem is nonlinear and nonconvex, and the solution seeks to minimize the total air power delivered by the fans in the network (Wu and Topuz 1998). The product of the fan pressure (Fj) and the volumetric flow (Qj) passing through the fan in the j th branch of the network determines the air power consumed by each fan, as Equation (11) presents. The sum of all the individual fan air powers determines the total air power consumed within a mine ventilation network. For branches that do not contain a fan, the fan pressure drop term is set to zero. The energy contributed by fans, as Equation (12) represents, serves to generate the required airflow and pressure distribution through the mine ventilation network.
The solution to the optimization problem identifies the number, location, and duty (or operational point, which is a pressure and airflow combination) of fans and regulators; these fans and regulators generate the airflow and pressure distribution to minimize the total fan air power or energy consumption of the mine, while delivering the airflow requirements at all the locations fixed by engineers.
Once the layout of a mine ventilation network is defined, a number of variants of the optimization problem may be solved. These variants may be categorized as (1)

Solution Methods
In recent years, several studies have explored the use of mathematical methods to minimize the power consumption of mine ventilation networks; these include Topuz (1989), Calizaya et al. (1987), Wang (1989), Barnes (1989), Jacques (1991), Huang and Wang (1993), Kumar et al. (1995), Wu and Topuz (1998) Two classes of methods were identified to solve the semi-controlled flow-splitting variant of the optimization problem. We describe the first method, developed by Calizaya et al. (1987), in this section. The second method consists of five iterative steps, which allow the evaluation of the air power consumed by the network, considering changes of the airflow quantities in each branch. The core of the method is a gradient technique. We present a small network consisting of six junctions and eight branches.
costs of a mine ventilation network, which include the investment costs associated with the purchase, installation, and maintenance of the fans and regulators, and (2) "further research work is therefore needed" (p. 360).
The optimization problem, which Calizaya et al. (1987), Kumar et al. (1995), and Lowndes et al. (2005) study, considers the type 1 semi-controlled flow (SC1) variant. All the proposed solution techniques use an iterative algorithm to determine the resistance values of the regulators needed in the network. These methods restrict regulators to a prespecified selection of branches, which may include airways leading to the working faces. The methods that Wang (1989), Barnes (1989), Jacques (1991), Huang and Wang (1993), and Wu and Topuz (1998) propose are applicable to the type 2 semi-controlled flow (SC2) variant of the optimization problem, in which a regulator may placed in any branch of the network. Acuña et al. (2010b) consider the free flow-splitting variant of the optimization problem, in which regulators are not permitted. Except for Wang (1989) and Acuña et al. (2010b), all the studies in Table 2 claim to achieve a local optimal solution for the variant of the PMVN optimization problem considered.
However, none of these studies provides proof of global optimality. Calizaya et al. (1987)  ventilation devices (i.e., fans or regulators) to achieve the desired airflow distribution. Barnes (1989) decomposes the type 2 semi-controlled flow variant of the optimization problem into two problems: (1) the determination of the airflow distribution problem, and (2) the evaluation of the pressure distribution required to generate the computed airflow distribution.
Barnes employs "a network adaptation of the classical nonlinear programming loop between a search for an improving direction, followed by a one dimensional optimization in the current improving direction" (p. 399). In simple terms, the method seeks an improvement solution search direction, followed by the calculation of the step in the current improvement direction. The algorithm starts with a possible set of flows and then incrementally improves the flows while preserving feasibility. The algorithm ends when no improvement direction is possible, as Barnes (1989) indicates. Jacques (1991) presents a heuristic optimization approach to compute the operational duties for ventilation devices (e.g., fans and regulators), which satisfy the required airflow distribution, to solve the type 2 semi-controlled flow variant of the optimization problem. The solution method can quickly respond to the daily changes in the demands placed on a real mine ventilation system. The heuristic concentrates on minimizing the airflow deviation from the requirements instead of minimizing the air power, as in other approaches. The technique defines the airflows that minimize the deviation from the airflow requirements, and then uses Kirchhoff's second law to determine the pressure required from fans and regulators available. As a result, the heuristic achieves the required airflow distribution by iteratively changing the duty points of fans and regulators. However, the method cannot locate fans and (or) regulators. Jacques (1991) indicates "when automatic control is aimed at, a method such as the one described here should be integrated as a module into an expert system software package which is destined in the future to take over from the ventilation officer" (p. 415). Huang and Wang (1993) propose an iterative approach to solve the type 2 semicontrolled flow variant of the optimization problem based on a generalized reduced gradient (GRG) method that employs multiple initial values. The objective function minimizes the total air power delivered by the installed fans. The authors provide a detailed description of the major computational steps involved in applying the GRG method, which are (1) the determination of a search direction, and (2) the calculation of the step in the improvement direction. "The relative error is used as the termination criterion" (p. 157), considering that "air quantity in a branch is usually within a hundred m3/s" (p. 157) and "fan or regulator pressure could be as high as a few thousand Pa" (p. 157). The presentation of the solution for a small-scale mine ventilation network model, consisting of 18 branches and nine nodes, illustrates the use of the method. Kumar et al. (1995) present a two-step algorithm to solve the type 1 semi-controlled flow variant of the optimization problem. The first step determines the best duty and location of the main surface fan(s) using the CPM. The CPM determines the largest pressure drop of the network, which is used to calculate the pressure value of the fan(s). The second step identifies the pressure duty of the underground booster fan(s), for predefined locations, using a four-stage heuristic approach that includes a Fibonacci and cyclic search routines (Bazaraa and Shetty 1979), coupled with the use of a steady state mine ventilation network program. The second phase of the algorithm determines a search direction and a step in the improvement direction.
The objective function minimizes the air power consumed by fans. The best solution identifies that only one of the branches with specified airflow does not need to have a regulator installed to achieve the required airflow distribution, which minimizes the energy consumed by regulators. Wu and Topuz (1998) propose a solution algorithm to the type 2 semi-controlled flow variant of the optimization problem. The algorithm employs a combination of the special ordered sets construct, the branch-and-bound algorithm, and linearization techniques, as Beale and Forrest (1976), Bertsekas (2003), and Luenberger and Ye (2008), respectively, discuss. The proposed algorithm transforms the initial nonlinear, nonconvex programming problem into a linear problem by introducing a special ordered set of variables. Then, the branch-and-bound method is used to explore the special ordered set of variables by directly partitioning the problem into subproblems, repeating the same procedure for each subproblem. Each subproblem is a linear programming model, because at that stage, the airflows have been fixed and the resulting problem is that of controlled flow splitting, which is linear. The resulting solution determines the fan pressures and regulator resistances necessary to obtain the required airflow distribution. Lowndes and Yang (2004) and Lowndes et al. (2005) develop and apply a genetic algorithm, coupled with a steady state mine ventilation network program, to solve the type 1 semi-controlled flow variant of the optimization problem. The resulting algorithm is tested with the primary mine ventilation system of the former Chilean El Indio mine. The genetic algorithm search is guided by a fitness function composed of the total consumed air power coupled to a penalty function employed to reject the solutions that do not deliver the necessary airflow at the specified airflow branches. The genetic algorithm discovers the location and pressure of underground booster fans needed to support a main fan in the delivery of the required airflow distribution. A commercial steady state mine ventilation network program determines the values of any regulator resistances installed in the specified airflow branches. The study concludes that the genetic algorithm identifies an optimal solution when 3 out of 16 underground booster fans are installed within the network. Acuña et al. (2010a) and Acuña et al. (2010b) extend the initial solution method proposed by Lowndes and Yang (2004) and Lowndes et al. (2005) to solve the free flow-splitting variant of the optimization problem, using a genetic algorithm coupled with a steady state mine ventilation network program.
The three solution methods, which Calizaya et al. (1987), Lowndes et al. (2005), and Acuña et al. (2010a)   We note that as the computational capabilities of computers have increased, the sizes of the problems that may be considered have increased from 2 to 3 fan locations and, at most, 20 branches (see Table 2), to sizes such as at the El Indio mine, which has 16 fan locations and 242 branches. Mines can have up to 20 primary mine ventilation fan locations and branches that number between 500 and 3000.
Only the studies presented by Lowndes et al. (2005) and Acuña et al. (2010b) include the computational times associated with each solution method; the other studies in Table 2 do not report it. We see a need to define an established set of representative benchmark mine ventilation networks against which the computational speed and efficiency of future solution methods may be compared.

Conclusions
The paper presents a review of the current state of the art of PMVN optimization methods.
Currently, mine ventilation engineers, guided by good practice and experience, employ existing mine ventilation network programs to determine the best mine ventilation configuration, using an iterative solution method to deliver the required airflow distribution at minimum cost. This  ηj = the efficiency of the fan installed in branch j (as a fraction between 0 and 1).
The decision variables are defined as:  Fj = the pressure developed by fan installed in branch j (kPa);  Qj = the airflow quantity flowing through branch j (m 3 /s);  reg j s = the shock-loss-equivalent length of the regulator in branch j (m).
The auxiliary variables are defined as:  Ej = the energy cost of the fan air power contribution into branch j ($USD);  Hj = the pressure drop of branch j (kPa);  Lj = the frictional pressure drop of branch j (kPa);  Pj = the fan air power contribution into branch j (kW, i.e., kNm/s);  * j r = the total shock-loss-equivalent resistance of the regulator in branch j (Ns 2 /m 8 );  Rj = the regulator pressure drop of branch j (kPa);  Z = the sum of fans air power of the mine (kW).