The Energy-Efficient Operation Problem of a Freight Train Considering Long-Distance Steep Downhill Sections

With the energy consumption rising in rail transport, the railway sector is showing increasing interest in the energy-efficient operation of freight trains. Freight trains require more complicated driving strategies than ordinary passenger trains do due to their heavy loads, especially in the long-distance steep downhill (LDSD) sections that are very common in freight rail lines in China. This paper studies the energy-efficient operation of a freight train considering LDSD sections. An optimal control model including regenerative and pneumatic braking is developed for the freight train. Then, when a train leaves/enters the LDSD section, we verify the uniqueness of control transitions and discuss the speed profile linkage between LDSD and its adjacent sections, which indicates that the periodic braking should be applied on LDSD sections for optimality. Additionally, given the same running time for the entire journey, our analysis shows that electrical braking-full braking strategy is more energy-efficient than coasting-full braking strategy on LDSD sections. Finally, a numerical algorithm for the optimal driving solution is proposed. The simulation results demonstrate that the driving strategies generated by the proposed algorithm performs better than those from fuzzy predictive control and field operation regarding energy saving.


Introduction
Freight trains play a major role in cargo transport all over the world.Due to the rising energy consumption, more attention has been paid to freight trains.Taking China as an example, the hauling energy consumption of locomotives accounts for 70%-75% of the total energy consumption of the railways [1].Thus, due to economic and environmental concerns it's very necessary to reduce the energy consumption of railways.
As we all know, the terrain in China is high in the west and descends toward the east coast.Thus, some rail lines consist of track sections where the train speed still increases even if full (maximum) electrical braking applied, i.e., long-distance steep downhill (LDSD) sections.The train driver must deal with the energy-efficient operation problem of the freight train in LDSD sections.Various methods have been adopted to reduce the energy consumption of freight trains, such as optimizing the timetable, changing the appearance of the locomotive to lower its aerodynamic resistance, installing regenerative brakes to feed the energy back to the traction network [2].Additionally, a good driving strategy is an economic method with great potential for reducing energy consumption, which is an interesting issue.

Previous Work for the Train Energy-Efficient Operation
Lots of methods have been utilized to solve the train energy-efficient operation problem since the 1960s [3,4].Reviewing the approaches, we can divide them into two categories: classic methods and intelligent methods.Classic methods refer to the optimization approaches that use precise mathematical models with rigorous derivations.Intelligent methods are the optimization approaches which solve the train energy-efficient operation problem using multiple objectives and constraints.
1.1.1.Classic Methods Maximum Principle (MP) was firstly employed in the train trajectory optimization by Ichikawa [4].Then the researchers from Scheduling and Control Group (SCG) improved and developed the application of MP to the problem of train trajectory optimization.Based on MP, three optimal controls for short-distance transport trains were presented by Milroy [5]: maximum power, coast and maximum brake.Asnis et al. [6] proposed that the optimal speed profile consists of four optimal controls: maximum power, cruise, coast and maximum brake.Lee et al. [7] noted that "cruise" was the fourth optimal control for long-distance transport trains.Howlett et al. [8] came up with a new local energy minimization principle to calculate the optimal switching points of the long-haul freight train along a single steep downhill or uphill track section.Albrecht et al. [9,10] discussed the existence and uniqueness of the train operation optimal strategy.The relationship between energy consumption and operation time was studied by Howlett [11].Meanwhile, other researchers also studied the train trajectory optimization based on MP.Khmelnitsky [12], Golovitcher and Iakov [13], and Liu [14] presented the necessary conditions of the train energy-efficient operation based on MP.Numerical algorithms were used to search for the optimal control strategy when the speed limit was considered in the optimal control problem of the train energy-efficient operation.Yang et al. [15] presented the optimal model of train operation with the regenerative braking considered.And the optimal solution from coasting to braking consists of a set of speed trajectories.
Combined with the Karush-Kuhn-Tucker (KKT) condition and derivative approach, the Lagrangian multipliers method was applied in the trajectory optimization of diesel locomotives with discrete controls.For a train along the level track within a given time, Jiaxin [16,17] noted that the energy consumption of the diesel locomotive was minimized if the velocity of the train was kept close to the critical velocity.Then Cheng's theory was improved by Howlett.The critical velocity was identified as the selected holding speed.The Lagrangian multipliers were introduced to search the energy-efficient driving strategies of a long-haul freight train with discrete control inputs.And the optimal strategy of a freight train on the level track is power-coasting-power between two stations [18,19].For a train on the continuously varying gradient, an iteration method was utilized to find optimal solution to the train energy-efficient operation [20].
With the improvement of computer processing speed, dynamic programming (DP) was applied to the train trajectory optimization.A nonlinear train optimal control problem considering the efficiency of the train traction system was solved via Discrete Dynamic Programming (DDP) [21,22].Miyatake et al. [23,24] used Bellman's DP to deal with optimization of the running profile of metro and electric trains.To overcome the dimension explosion of DP, sequential quadratic programming (SQP) was applied to the trajectory optimization of a railway vehicle [25,26].
The calculus of variation is also used to solve the train's minimal-energy driving strategy problem.There were two levels in the train control model: discrete control, i.e., the optimal controls such as power, coasting and braking; continuous control such as the traction force varying with the train velocity.A hybrid automata was used to describe the hierarchical control model [2,27].
Mixed-integer linear programming (MILP) that was appropriate for the engineering application was utilized in an automatic train operation (ATO) system to obtain the optimal control sets [28].The gauss pseudospectral method was applied in the single and multiple trains trajectory optimization [28,29], which was a new approach in the train trajectory optimization in recent years.

. Intelligent Methods
Recently many intelligent approaches have been applied to the train trajectory optimization problem, which have the advantage of providing rapid solution to the Multi-Input Multi-Output (MIMO) problems with multiple objectives and constraints.For example, Genetic Algorithm (GA) was used to search for the optimal train speed profile [30].Based on the necessary conditions derived from MP, GA was proposed to search for the proper coasting points to realize the energy-efficient operation of a train [31].Zhao et al. [32] presented a multiple train trajectory optimization based on GA to make a compromise between energy use and punctuality when delays occurred.Huang et al. [33] proposed the Multi-population Genetic Algorithm (MPGA) to solve the train energy-efficient operation problem by optimizing the train operation for multiple interstations.Meanwhile, fuzzy predictive control (FPC) was applied to the ATO by Feng [34].Bai et al. [35] employed FPC to optimize the trajectory of freight trains with diesel locomotives.Ant colony optimization (ACO) was also applied to obtain the single train optimal speed profile [30].

Main Contribution
Train trajectory optimization was studied using various approaches in the previous work .Meanwhile, references [8][9][10][11] studied the optimal controls of the freight train along the single steep downhill track section, which is not effective for the long steep downhill sections.In brief, the trajectory optimization of the train operation considering the LDSD hasn't been discussed yet.In this paper, the abovementioned problem will be worked out.
This paper provides four main contributions to the literature.Firstly, the mathematical model of the train energy-efficient operation problem considering LDSD has been established.Then the analytical solution is derived.Secondly, the pneumatic braking (or mechanical braking) force is separated from the integration braking force (including electrical braking and pneumatic braking) while deriving the necessary conditions for the energy-efficient operation of the train.Analysis of the adjoint variable shows that the periodic braking strategy should be applied on the LDSD section.Thirdly, we transform the time constraint with respect to the auxiliary reservoir air-filled time (see Section 2.3) to the speed constraint, which could be considered the path constraint in the optimal control problem.Fourthly, the optimal control at the terminal position of the LDSD section is verified.
Although we obtain the analytical solution to the train speed and optimal control, it's not enough to calculate the optimization trajectory.Based on the above conclusions, we give the linkage rules of the speed holding section incorporating LDSD.Then an iterative numerical algorithm is proposed to obtain the optimal speed profile.The comparative simulation results show that the proposed algorithm in this paper saves more energy, which can provide the train driver the significant guidance with respect to the energy-efficient operation issue.

Main Structure of this Paper
This paper is organized as follows: Section 2 proposes the motion model of a freight train with its constraints, which is considered as an optimal control problem; Sections 3 and 4 give the way to solve the aforementioned optimal control problem; Section 5 analyzes the influence of the journey time on the energy consumption between stations; in Section 6, the numerical algorithm is proposed to solve the optimal control problem; Section 7 illustrates the algorithm in the case studies; in Section 8, conclusions and future work are given.

Real Characteristics of a Freight Train
Electric locomotives such as China railways HXD1 and HXD2 are utilized for freight transport in the Chinese main line railway sector.If the traction force at low speed stage is set too large, the motor in the locomotive will idle, which would cause wheel slippage.The electrical braking force is similar.
Here, an HXD2 is chosen as the simulation locomotive, which is shown in Figure 1.The HXD2 that is the source power of a freight train is driven on the track by the adhesion traction force.v loc is the linear velocity of the wheel in the locomotive.v w is the angular velocity of the wheel in the locomotive, and v loc = r • v w , where r is the radius of the wheel.N is the opposite force from the track and f c (v) is the traction force generated by the motor.f r (v) is the rolling friction.Here, an HXD2 is chosen as the simulation locomotive, which is shown in Figure 1.The HXD2 that is the source power of a freight train is driven on the track by the adhesion traction force.loc v is the linear velocity of the wheel in the locomotive.The rolling friction is caused by the motion of the wheels of the locomotive.Assume that A is the contact point between the wheel and the track.Since the motion direction of A is opposite to that of the locomotive, then () r fv occurs to prevent the motion of A as follows: where N is equal to the weight of the locomotive, j  is the rolling friction coefficient of the wheel.
It is necessary to point out that j  is from a number of field tests [36], i.e., 10 0.11 0.38 10 0.2 where v is equal to loc v .When a locomotive is hauling, j  is the rolling friction coefficient.
In Figure 2a, () fv is the traction force of the locomotive (blue solid line) and () r fv is the adhesion traction force of the locomotive (red dash line), which is divided into two parts.The first part (the traction force) that varies linearly with the speed 65 / t v km h  is called the "constant torque zone", while the second part (the traction force) that is inversely proportional to the speed 65 / t v km h  is called the "constant power zone".Thus, the adhesion traction force varies nonlinearly with the speed.During the low speed stage, the real traction force of a locomotive is less than () c fv owing to the restriction of the adhesion force.At the high speed stage, the real traction force of a locomotive is equal to () c fv.In short, it can be expressed as: where 0 () t fv is the real traction force of a freight train.The rolling friction is caused by the motion of the wheels of the locomotive.Assume that A is the contact point between the wheel and the track.Since the motion direction of A is opposite to that of the locomotive, then f r (v) occurs to prevent the motion of A as follows: where N is equal to the weight of the locomotive, µ j is the rolling friction coefficient of the wheel.It is necessary to point out that µ j is from a number of field tests [36], i.e., where v is equal to v loc .When a locomotive is hauling, µ j is the rolling friction coefficient.
In Figure 2a, f c (v) is the traction force of the locomotive (blue solid line) and f r (v) is the adhesion traction force of the locomotive (red dash line), which is divided into two parts.The first part (the traction force) that varies linearly with the speed v t < 65 km/h is called the "constant torque zone", while the second part (the traction force) that is inversely proportional to the speed v t ≥ 65 km/h is called the "constant power zone".Thus, the adhesion traction force varies nonlinearly with the speed.During the low speed stage, the real traction force of a locomotive is less than f c (v) owing to the restriction of the adhesion force.At the high speed stage, the real traction force of a locomotive is equal to f c (v).In short, it can be expressed as: where f t0 (v) is the real traction force of a freight train.Similar to Equation (2), when a locomotive is using electrical braking, the rolling friction coefficient µ z is as follows [36]: where v is equal to v loc .The rolling friction coefficient is: In Figure 2b, the braking force of the HXD2 locomotive is defined as f eb (v), i.e., regenerative braking force (yellow solid line).When the electrical braking is applied, the motor in the locomotive will generate electricity.As a result, the electricity is fed back to the traction network so that other trains can utilize it.There are two stages in the electrical braking characteristics.The first changes linearly with the speed v b < 70.6 km/h , while the latter changes nonlinearly with the speed v b ≥ 70.6 km/h , i.e., inversely proportional to the speed.f rz (v) that is the adhesion braking force of the locomotive (red dash line) changes nonlinearly with the velocity of the locomotive determined by (5).Thus, the real braking force of the locomotive can be obtained: where f d0 (v) is the real regenerative braking force.Similar to Equation (2), when a locomotive is using electrical braking, the rolling friction coefficient z  is as follows [36]: where v is equal to loc v .The rolling friction coefficient is: In Figure 2b, the braking force of the HXD2 locomotive is defined as () eb fv , i.e., regenerative braking force (yellow solid line).When the electrical braking is applied, the motor in the locomotive will generate electricity.As a result, the electricity is fed back to the traction network so that other trains can utilize it.There are two stages in the electrical braking characteristics.The first changes linearly with the speed 70.
fv that is the adhesion braking force of the locomotive (red dash line) changes nonlinearly with the velocity of the locomotive determined by (5).Thus, the real braking force of the locomotive can be obtained: where 0 () d fv is the real regenerative braking force.For a freight train, there are two types of braking force.One is the regenerative braking force.The other is mechanical braking force, which is illustrated in [37].The mechanical braking force is presented as follows (see Figure 3).For a freight train, there are two types of braking force.One is the regenerative braking force.The other is mechanical braking force, which is illustrated in [37].The mechanical braking force is presented as follows (see Figure 3).Similar to Equation (2), when a locomotive is using electrical braking, the rolling friction coefficient z  is as follows [36]: where v is equal to loc v .The rolling friction coefficient is: In Figure 2b, the braking force of the HXD2 locomotive is defined as () eb fv , i.e., regenerative braking force (yellow solid line).When the electrical braking is applied, the motor in the locomotive will generate electricity.As a result, the electricity is fed back to the traction network so that other trains can utilize it.There are two stages in the electrical braking characteristics.The first changes linearly with the speed 70.
fv that is the adhesion braking force of the locomotive (red dash line) changes nonlinearly with the velocity of the locomotive determined by (5).Thus, the real braking force of the locomotive can be obtained: where 0 () d fv is the real regenerative braking force.
For a freight train, there are two types of braking force.One is the regenerative braking force.The other is mechanical braking force, which is illustrated in [37].The mechanical braking force is presented as follows (see Figure 3).In Figure 3, the adhesion force is greater than the mechanical braking force.For a freight train along the track, the mechanical braking force isn't influenced by the adhesion force.The real mechanical braking force can be described [36] using the following expression: Energies 2017, 10, 794 where β c is the service braking coefficient (equal to 0.17 in this paper).K h is the force of the brake shoe.
For a locomotive, it's 580 kN.For a fully loaded wagon, it's 195 kN.ϕ h is the friction coefficient, which varies nonlinearly with the speed of the train.For a locomotive [36], we have: For a fully loaded wagon [36], we have:

Train Model
The freight train model is proposed in this section.Every wagon in the freight train can possibly run on a section of track with a different gradient.For an example, the front wagon may run on an uphill section while the middle one is on a downhill section.Therefore, a multi-particle model of a freight train is more appropriate in this paper instead of a single-particle one.The train marshalling with one locomotive and n − 1 wagons is shown in Figure 4.
In Figure 3, the adhesion force is greater than the mechanical braking force.For a freight train along the track, the mechanical braking force isn't influenced by the adhesion force.The real mechanical braking force can be described [36] using the following expression: where c  is the service braking coefficient (equal to 0.17 in this paper).
h K is the force of the brake shoe.For a locomotive, it's 580 kN.For a fully loaded wagon, it's 195 kN.h  is the friction coefficient, which varies nonlinearly with the speed of the train.For a locomotive [36], we have: For a fully loaded wagon [36], we have:

Train Model
The freight train model is proposed in this section.Every wagon in the freight train can possibly run on a section of track with a different gradient.For an example, the front wagon may run on an uphill section while the middle one is on a downhill section.Therefore, a multi-particle model of a freight train is more appropriate in this paper instead of a single-particle one.The train marshalling with one locomotive and n − 1 wagons is shown in Figure 4. Due to the Figure 4 and considering the position of the train (x) as an independent variable, the dynamic equations are as follows: ) where x is the position of the train, and vx is the velocity of the train at x, and () tx is the journey time of the train at x.The control variables: x are the unit traction and electrical braking force of all locomotives in the freight train (N/kg), respectively.When M is considered as the mass of the freight train, we have m f v x is the unit mechanical braking force of the freight train (N/t) with the relationship is the real mechanical braking force of the ith wagon or locomotive.0 ( ( )) w v x (N/t) that is the train unit basic resistance consists of rolling resistance and aero-dynamic resistance. ()gx is made up of two components: unit gradient resistance () grad gx (N/t) and unit curve resistance () curve gx (N/t).0 ( ( )) w v x , () gx can be described as: Due to the Figure 4 and considering the position of the train (x) as an independent variable, the dynamic equations are as follows: where x is the position of the train, and x = x 1 , v(x) is the velocity of the train at x, and t(x) is the journey time of the train at x.The control variables: ) are the unit traction and electrical braking force of all locomotives in the freight train (N/kg), respectively.When M is considered as the mass of the freight train, we have real mechanical braking force of the ith wagon or locomotive.w 0 (v(x))(N/t) that is the train unit basic resistance consists of rolling resistance and aero-dynamic resistance.g(x) is made up of two components: unit gradient resistance g grad (x)(N/t) and unit curve resistance g curve (x)(N/t).w 0 (v(x)), g(x) can be described as: where a, b and c are positive constant coefficients, and the equivalent unit gradient resistance and curve resistance are mass of ith wagon or locomotive.For simplicity, there is v = v(x).
Remark 1.The control variable of pneumatic braking is binary, which is different from the continuous control variable of traction and electrical braking.Theoretically, the control variable of pneumatic braking varies continuously with the pressure reduction of the train brake pipe.However, the pressure reduction is discretized based on the operation rules of the freight train in China.The train driver is required to apply the discrete pressure reduction to the freight train.Hence, while modelling the freight train, the discrete pressure reduction is taken into account.And the pneumatic braking force with minimum pressure reduction could make the train speed decrease.Then the binary control variable is introduced-µ m ∈ {0, 1}.If µ m = 0, the pneumatic braking is not applied.If µ m = 1, the pneumatic braking is applied.

Problem Formulation
Let 0 and X be the initial and terminal positions of the freight train.The velocity of the freight train should be 0 at the origin and destination station: Assume the journey time from 0 to X is T. The constraint is: Exceeding the line speed limit is not allowed for safety concerns: where V(x) is the static speed limit at position x.
The pressure in the freight train brake pipe increases while the mechanical braking is released [37].It takes time to recharge the brake pipe to the target pressure (600 kPa).The time is called the auxiliary reservoir air-filled time.To ensure enough braking force is available in the next braking period, the air-filled time should not be greater than the speed rising time [38]: where t z and T c are the speed rising time and the auxiliary reservoir air-filled time, respectively.Considering the LDSD section, the energy-efficient operation problem of a freight train that is to find an optimal train trajectory minimizes the energy consumption.Due to the principle of the optimal control, a cost function is defined as follows: where η t is the energy efficiency ratio of the traction system, η d is the energy efficiency ratio of the electrical braking system, α is the regeneration efficiency representing the feedback energy percentage of the electrical braking.
Remark 2. The aim of this paper is to minimize the energy consumption of the traction system (including power and regenerative braking).Our aim is not the same as minimizing total energy consumption of the traction power supply system because the energy efficiency ratios vary with the train speed and the loads, i.e., power and electrical braking.In practice, we could try to turn off some motors in the locomotive at the cruising phase to make the traction system run near the peak efficiency at high load.Then the energy efficiency ratios could be approximatively considered as a constant.
Remark 3. The regeneration efficiency is introduced for two purposes while modelling the freight train, which is illustrated below: (1) In most electric trains, the regenerative braking energy can be fed back to the traction power supply system if other trains along the railway line could utilize it.Currently, freight trains are not equipped with on-board energy storage devices.Hence, the regeneration efficiency is decided by the availability of other trains.In the realistic traction power supply system, the regeneration efficiency is a variable.The railway operation scheme (including the timetable and number of freight trains that are running along the railway line) should be investigated to obtain the accurate regeneration efficiency.Based on the operation data in months or years, we could calculate the regeneration efficiency.In China, the railway electrification system is 2 × 25 kV alternating current (AC) single phase with a frequency of 50 Hz.Two phases of the three-phase AC power from the public power grid are used to supply for the electric trains.The phases at different feeders of the substations will be switched between A-C, C-B, B-A, to maintain the overall balance of the public power grid (see Figure 5).Since the double-track railway including up rail and down rail is very common in China, the traction network of up rail and down rail connects in parallel.We can optimize the timetable to make the up train and down train run on the same LDSD section.For example, for a fully loaded train that is running on the up rail of the LDSD section, if electrical braking is applied, the electricity generated by the up train will be fed back to the traction network.At the same time, if the down train with empty wagons is going uphill with power on the LDSD section or the fully loaded up train is powering on the normal gradient (see Figure 5), α of the energy generated by up train will be absorbed by the down train.
(2) In this paper, our aim is to calculate the optimal speed trajectory considering the LDSD section.Our work in this paper is the operation optimization for a single train.The regeneration efficiency is introduced into the optimal control problem to evaluate the ratio of coasting and regenerative braking.To simplify the optimal control problem in this paper, the regenerative efficiency is considered as a constant.
Energies 2017, 10, 794 8 of 26 Remark 3. The regeneration efficiency is introduced for two purposes while modelling the freight train, which is illustrated below: (1) In most electric trains, the regenerative braking energy can be fed back to the traction power supply system if other trains along the railway line could utilize it.Currently, freight trains are not equipped with onboard energy storage devices.Hence, the regeneration efficiency is decided by the availability of other trains.In the realistic traction power supply system, the regeneration efficiency is a variable.The railway operation scheme (including the timetable and number of freight trains that are running along the railway line) should be investigated to obtain the accurate regeneration efficiency.Based on the operation data in months or years, we could calculate the regeneration efficiency.In China, the railway electrification system is 2 × 25 kV alternating current (AC) single phase with a frequency of 50 Hz.Two phases of the threephase AC power from the public power grid are used to supply for the electric trains.The phases at different feeders of the substations will be switched between A-C, C-B, B-A, to maintain the overall balance of the public power grid (see Figure 5).Since the double-track railway including up rail and down rail is very common in China, the traction network of up rail and down rail connects in parallel.We can optimize the timetable to make the up train and down train run on the same LDSD section.For example, for a fully loaded train that is running on the up rail of the LDSD section, if electrical braking is applied, the electricity generated by the up train will be fed back to the traction network.At the same time, if the down train with empty wagons is going uphill with power on the LDSD section or the fully loaded up train is powering on the normal gradient (see Figure 5),  of the energy generated by up train will be absorbed by the down train.(2) In this paper, our aim is to calculate the optimal speed trajectory considering the LDSD section.Our work in this paper is the operation optimization for a single train.The regeneration efficiency is introduced into the optimal control problem to evaluate the ratio of coasting and regenerative braking.To simplify the optimal control problem in this paper, the regenerative efficiency is considered as a constant.The problem can be formulated as follows: Find a series of optimal controls and switching locations to make the cost function (18) minimized.Meanwhile, the dynamic constraints (10) and (11), the path constraints ( 16), (17), the boundary constraints ( 14) and (15), must be satisfied.

Solution
Section 2 gives the formula of the train energy-efficient operation problem.This section discusses the analytical solution to the train trajectory optimization.

Necessary Condition of the Train Energy-Efficient Operation
Combining (10), (11) and ( 18), the Hamiltonian is as follows: The problem can be formulated as follows: Find a series of optimal controls and switching locations to make the cost function (18) minimized.Meanwhile, the dynamic constraints (10) and (11), the path constraints ( 16), (17), the boundary constraints ( 14) and (15), must be satisfied.

Solution
Section 2 gives the formula of the train energy-efficient operation problem.This section discusses the analytical solution to the train trajectory optimization.
To simplify (19), a new adjoint variable is introduced: which converts Equation (19) as follows: To maximize (21), a set of six optimal controls is presented:

Transformation of the Time Constraint
In this subsection, a theorem about the air-fill time is given.Before that, the necessary definitions and assumption are given.
Definition: Let v (x) be the instantaneous speed from A to B. Let the position of A and B be x A and x B .Let the position of C and D be x C and x D .T c is equal to the integration from x A to x B : where v (x A ) = ϕ(x A ) and v (x B ) = V(x).Assumption: suppose v(x) ≤ ϕ(x) does not hold, then, there is v(x) > ϕ(x) to assure the existing of the optimal solution.
The Lipschitz continuity should be satisfied: where L ≥ 0.
Theorem 1.Let ϕ(x) be the optimal release speed while the mechanical braking is removed.Let v(x) be the velocity of the train speed.ϕ and v is the function of the train position x.If and only if v(x) ≤ ϕ(x), the optimal solution to the problem exists.And T c equals to the time from A to B (see Figure 6) by applying full electrical braking (FEB).
Proof of Theorem 1. Proof by contradiction is presented as follows: Depending on the assumption, ∃ϕ (x) and ϕ (x) > ϕ(x), i.e., v(x A ) = ϕ (x A ). ϕ(x) and ϕ (x) are constants.To make the time from v(x A ) to v(x D ) equal to T c , there must be a point satisfying: Energies 2017, 10, 794 10 of 26 In Figure 6, the instantaneous speed v(x) is always greater than v (x) from x A to x C : which converts Equation ( 10) into the following expression: where v(x C ) and x C are the initial states of the differential Equation (27 In Figure 6, the instantaneous speed () vx is always greater than () vx  from A x to C x : , there is: which converts Equation (10) into the following expression: where () C vx and C x are the initial states of the differential Equation ( 27), i.e., 12 , ( ) 0  , given the initial condition: Equation ( 26) and

only one solution exists at [ , ]
CC xx   .However, Equation (26) implies that there are two solutions when C xx  .This contradicts the Lipschitz continuity (existence and uniqueness theorem of the differential equation).Then we have: , there is: . Meanwhile: and the assumption does not hold.If and , Theorem 1 holds.Finally, the time constraint is transformed to the speed However, Equation (26) implies that there are two solutions when x > x C .This contradicts the Lipschitz continuity (existence and uniqueness theorem of the differential equation).Then we have: which implies v(x) ≥ v (x).Meanwhile: Obviously, Equation ( 29) does not hold if v(x) ≥ v (x) and the assumption does not hold.If and only if v(x) ≤ ϕ(x), Theorem 1 holds.Finally, the time constraint is transformed to the speed constraint.

Solution to the Optimal Control Problem with the Speed Constraint
The necessary condition of the freight train energy-efficient operation is given in Section 3.1.Further study is proposed to solve the problem with the speed constraint.According to Equation (15) and Theorem 1, two complementary slackness factors (M(x) and T(x)) are introduced [14,38]: where The derivative of the Lagrange Multipliers considering the slackness factors is: In (31), the Hamiltonian does not depend on t explicitly, the partial derivative of H is zero: thus, λ 1 = C, where C is a constant.Equation ( 20) introduces an adjoint variable to replace the Lagrange Multiplier λ 2 .Differentiating (20) yields the following: which can be rewritten as: Combining ( 19), ( 32) and ( 35): In Section 3.1, only the general relationship between the adjoint variable and the train optimal controls is derived, i.e., the necessary condition.However, it's not enough to calculate the precise speed trajectory of the freight train energy-efficient operation.In this section, Equation (36) of the adjoint variable is obtained to determine the optimal controls for the train operation.

Analysis of the Special Optimal Controls
To make the freight train operation energy-efficient, there are six optional optimal controls.Among them, there are three unique optimal controls, i.e., PP, PEB and FB.We will discuss the special optimal controls to get more details about the train energy-efficient operation.
To find those, a speed function is defined as follows: Due to Equations ( 12) and ( 40) can be rewritten as: Obviously, ζ(v) is monotone increasing for ζ (v) > 0. When θ = 1, only one optimal constant speed v c that makes Equation (39) hold.
Let the target speed of the train operation be v t .When θ = 1, there is: The mathematical description of the normal gradient where a freight train can maintain a constant speed is presented: when the freight train runs on the interval with partial electrical braking (PEB), we have the adjoint variable θ = αη d η t , and its corresponding optimal control: 36) with θ = αη d η t yields the following: Similarly, there is: which can be rewritten as: when a freight train is using partial electrical braking, there is only one optimal constant electrical braking speed for Equation (46) due to the monotone increasing of ζ(v).
Let the target speed of the train operation be v td .When θ = αη d η t , there is: The mathematical description of other type of the normal gradient where a freight train can maintain a constant speed is presented:

Uniqueness Analysis of the Optimal Control at the Terminal Position of the LDSD Section
Since the freight train operation on a LDSD section is different from that on a normal gradient, the driving strategies that are utilized on the normal gradient can't be used to get through the LDSD section.It's necessary to analyze the optimal controls for the LDSD section.
In this section, a theorem about the uniqueness of the optimal control on the LDSD section will be proved, which gives out the optimal control at the terminal position of the LDSD.
Theorem 2. Assume that a freight train is at the terminal position of the LDSD (see Figure 6).While v c > V(x), we obtain that the adjoint variable is 0 < θ < αη d η t , and the optimal control is FEB, the velocity of the train is equal to the line speed limit.

Proof of Theorem 2. Proof by contradiction:
Assumption: the optimal control at the terminal position of the LDSD is not FEB.Then the possible optimal controls are FP, PP, PEB, C or FB.The comparison of FEB and other optimal controls is presented in Figure 7.In Figure 7a-d, ϕ(x) is the pneumatic braking release speed by applying "FB+FEB".

• FP
If the freight train is accelerated to V(x) with FP (see Figure 7a), the air-filled time holds on via Theorem 1.Then, there is v(x) ≤ ϕ a (x), where ϕ a (x) is the pneumatic braking release speed by applying "FB+FP".Since the acceleration of FP is greater than that of FEB, ϕ a (x) is far less than ϕ(x) so that there is sufficient air in the auxiliary reservoir for the next pneumatic braking.Therefore, it's very possible that ϕ a (x) is less than 30 km/h [39], which will cause large longitudinal impact on the wagon couplers.Meanwhile, the freight train is running on the energy-efficient slope, i.e., LDSD section.It does not necessarily consume energy from the traction network.In summary, FP is not the optimal control at the terminal position of the LDSD.
• PP Equation ( 38) can be rewritten as: In Figure 7b, ϕ b (x) is assumed as the holding speed or pneumatic braking release speed with the relationship ϕ b (x) < v c .If the trip time is given, Section 3.4.1 implies that there is only one optimal holding speed v c to make dθ/dx = 0.Meanwhile, ζ(v) is monotone increasing.When ϕ b (x) < v c , there is dθ/dx < 0, i.e., θ is monotone decreasing.
For a freight train that is running at x 1 with PP (v(x 1 ) = ϕ b (x 1 )), the corresponding adjoint variable θ(x 1 ) = 1.If the train runs from x 1 to x 2 with PP, we have v( Once θ is monotone decreasing, then the adjoint variable θ(x 2 ) < 1. Obviously, the optimal control for the freight train at x 2 is not PP.However, this contradicts with the assumption: the train should be running on the LDSD with PP after pneumatic braking is released.Thus, PP is not the optimal control.

• PEB
In a similar way, it can be proved that the velocity of the train is not maintained if PEB is applied on the LDSD section, which contradicts with the assumption-the train should be running at constant speed v d based on Section 3.4.2.Then, PEB is excluded.

• C
According to Equation ( 18), there is minJ 1 = 0 at the coasting phase (see Figure 7c).On the other hand, while FEB is applied, the energy consumption is: where X 2 is the running distance by applying FEB.It is clear that minJ 2 < minJ 1 .Thus, it's more energy-efficient to apply FEB.Meanwhile, if FEB is applied, the electricity generated from the motor in the locomotive can be fed back to the traction network so that other trains absorb it.If C is applied, the pneumatic braking speed ϕ c (x) that may be very low leads to large longitudinal impact on the couplers.Thus, C is not the optimal control.

• FB
Assume that ϕ d (x) is the pneumatic braking release speed by applying "FEB+FB" (see the blue line in Figure 7(d)).We have ϕ(x) = ϕ d (x).Compared with the energy consumption of accelerating the train speed from v c to V(x), more traction energy is consumed if the train speed is accelerated from v c to ϕ d (x) (the blue line).Therefore, FB is not the optimal control.In summary, the assumption does not hold.The only optimal control is FEB.Theorem 2 states uniqueness of the optimal control at the terminal position of the LDSD.


(the blue line).Therefore, FB is not the optimal control.In summary, the assumption does not hold.The only optimal control is FEB.Theorem 2 states uniqueness of the optimal control at the terminal position of the LDSD.

Direction of the Linkage
Let the speed holding section be the interval with PP or PEB.The corresponding velocities are c v and d v .The optimization trajectory from the initial position ( 0 ) to the terminal position ( X ) is divided into three parts.Figure 8 shows the linkage of the three different parts.
As Theorem 1 and Theorem 2 should be satisfied in the optimization trajectory calculation, a

Direction of the Linkage
Let the speed holding section be the interval with PP or PEB.The corresponding velocities are v c and v d .The optimization trajectory from the initial position (0) to the terminal position (X) is divided into three parts.Figure 8 shows the linkage of the three different parts.
As Theorem 1 and Theorem 2 should be satisfied in the optimization trajectory calculation, a backward integration (profile 1) is adopted to calculate the speed profile in the LDSD section.And a forward integration is adopted to calculate the speed profile in the normal gradient section.The two profiles meet at the switching position x 1 (see Figure 8), while the adjoint variables of the two speed profiles also meet at x 1 .
Meanwhile, another backward integration (profile 2) is adopted to calculate the speed profile from the terminal position (X) to the end position of the LDSD section.Backward profile 1 and backward profile 2 meet at the switching position x 2 .And the adjoint variables of the two speed profiles also meet at x 2 .
Energies 2017, 10, 794 15 of 26 Meanwhile, another backward integration (profile 2) is adopted to calculate the speed profile from the terminal position ( X ) to the end position of the LDSD section.Backward profile 1 and backward profile 2 meet at the switching position 2 x .And the adjoint variables of the two speed profiles also meet at 2 x .

Sufficient Conditions of State Variable Inequality Constraint
Seierstad and Sydsaeter [40] proposed a sufficient theorem to deal with the state variable inequality constraint (Equations ( 16) and ( 17)).Since  is the function of the train position x , a piecewise continuous function   12 ( ) ( ), ( ), , ( ) is assumed to be existed on the span [0, ] X which has a piecewise continuous derivative − / d dx  (see Equation ( 36)) except for a finite number of discontinuous points.At the points of discontinuity 12 , the following jump condition is satisfied: where: According to the Equation ( 30), the jump condition of  are as follows: (1) While ( )

Linkage Case
Two typical cases are used to illustrate the linkage of the speed holding section incorporating the LDSD section, i.e., () .  is the function of the train position x .The

Sufficient Conditions of State Variable Inequality Constraint
Seierstad and Sydsaeter [40] proposed a sufficient theorem to deal with the state variable inequality constraint (Equations ( 16) and ( 17)).Since θ is the function of the train position x, a piecewise continuous function θ ] is assumed to be existed on the span [0, X] which has a piecewise continuous derivative −dθ/dx (see Equation ( 36)) except for a finite number of discontinuous points.At the points of discontinuity τ 1 , the following jump condition is satisfied: where: • τ − i and τ + i denote the left-hand side and the right-hand side of the train position τ i .

Linkage Case
Two typical cases are used to illustrate the linkage of the speed holding section incorporating the LDSD section, i.e., v c > V(x) and v c < V(x).θ is the function of the train position x.The relationship of θ and x is presented.

Case 1
(1) In Figure 9, the optimal control of the freight train switches from PP to C before entering the LDSD section.Based on the linkage direction in Section 4.1, the intersection of forward profile (C) and backward profile 1 (FEB) is x = x 0 .(2) Entering the LDSD section, the speed of the freight train keeps increasing even if FEB is applied.
The adjoint variable (θ) is still decreasing.While θ = 0 − , v(x) = V(x).Although the jump condition of θ is satisfied, the jump will not happen yet due to the optimal controls.At the next moment, θ = 0 + , FB is applied.We have v(x) < V(x).
(3) Once FB is applied, pneumatic braking is released until v(x) = ϕ(x).Then dT(x)/dx > 0, θ jumps at τ − 1 .We have 0 < θ(τ + 1 ) < αη d η t , then FEB is applied.And FEB is kept until v(x) = V(x) and θ = 0 − .Then FB is applied again.(4) While the freight train is running at the terminal position of the LDSD section, we have Then FP is applied while the freight train is leaving the LDSD.Finally, the freight train is accelerated to v c .
Energies 2017, 10, 794 16 of 26 (2) Entering the LDSD section, the speed of the freight train keeps increasing even if FEB is applied.
The adjoint variable (  ) is still decreasing.While Although the jump condition of  is satisfied, the jump will not happen yet due to the optimal controls.At the next moment, 0    , FB is applied.We have ( ) ( ) (3) Once FB is applied, pneumatic braking is released until ( ) ( )  .Then FB is applied again.
(4) While the freight train is running at the terminal position of the LDSD section, we have

( ) ( )
vV    .For ( ) / 0 Then FP is applied while the freight train is leaving the LDSD.Finally, the freight train is accelerated to Linkage of the speed holding section considering the LDSD section ( ()

Case 2
(1) In Figure 10, the optimal control of the freight train switches from PP to C before entering the LDSD section.Based on the linkage direction in Section 4.1, the intersection of forward profile (C) and backward profile 1 (FEB) is (3) Once FB is applied, pneumatic braking is released until ( ) ( )  .Then FB is applied again.
(4) While the freight train is running at the terminal position of the LDSD section, we have

Case 2
(1) In Figure 10, the optimal control of the freight train switches from PP to C before entering the LDSD section.Based on the linkage direction in Section 4.1, the intersection of forward profile (C) and backward profile 1 (3) Once FB is applied, pneumatic braking is released until v(x) = ϕ(x).Then dT(x)/dx > 0, θ jumps at τ − 1 .We have 0 < θ(τ + 1 ) < αη d η t , FEB is applied.And FEB is kept until v(x) = V(x) and θ = 0 − .Then FB is applied again.(4) While the freight train is running at the terminal position of the LDSD section, we have Then FP is applied while the freight train is leaving the LDSD.Finally, the freight train will coast to v d .And the optimal control between PEB section and PP section is C.
. Linkage of the speed holding section considering the LDSD section ( ()

Relationship between Journey Time and Energy Consumption
In the previous articles [2,11,41], the conflicted relationship between the journey time and energy consumption was studied.From the perspective of traction energy consumption, we draw a conclusion: the total energy consumed (E) is a rigorously monotone decreasing function of the given journey time ( 0 T ).The relationship is presented in the following mathematical equation:    .The relationship can be illustrated in Figure 11.The conclusion will be adopted to analyze the relationship of the interval time allocation and the freight train's energy consumption.Depending on Theorem 2 and Section 4.3, the optimal driving strategy on the LDSD section should be the periodic braking, i.e., FEB+FB+FEB (see Figure 12), which increases the number of freight trains that pass Interval 2 and this decreases the total energy consumption from 0 to X.

Relationship between Journey Time and Energy Consumption
In the previous articles [2,11,41], the conflicted relationship between the journey time and energy consumption was studied.From the perspective of traction energy consumption, we draw a conclusion: the total energy consumed (E) is a rigorously monotone decreasing function of the given journey time (T 0 ).The relationship is presented in the following mathematical equation: where ψ (T 0 ) < 0. The relationship can be illustrated in Figure 11.

Relationship between Journey Time and Energy Consumption
In the previous articles [2,11,41], the conflicted relationship between the journey time and energy consumption was studied.From the perspective of traction energy consumption, we draw a conclusion: the total energy consumed (E) is a rigorously monotone decreasing function of the given journey time ( 0 T ).The relationship is presented in the following mathematical equation:    .The relationship can be illustrated in Figure 11.The conclusion will be adopted to analyze the relationship of the interval time allocation and the freight train's energy consumption.Depending on Theorem 2 and Section 4.3, the optimal driving strategy on the LDSD section should be the periodic braking, i.e., FEB+FB+FEB (see Figure 12), which increases the number of freight trains that pass Interval 2 and this decreases the total energy consumption from 0 to X.The conclusion will be adopted to analyze the relationship of the interval time allocation and the freight train's energy consumption.Depending on Theorem 2 and Section 4.3, the optimal driving strategy on the LDSD section should be the periodic braking, i.e., FEB+FB+FEB (see Figure 12), which increases the number of freight trains that pass Interval 2 and this decreases the total energy consumption from 0 to X. Analysis: in Figure 12, suppose there are two driving strategies on the LDSD section.
(1) Strategy 1: FEB+FB+FEB, the pneumatic braking release speed is equal to (2) Strategy 2: C+FB+C, the pneumatic braking release speed is equal to The trip from 0 to X is divided into three intervals as follows.According to the relationship between the energy consumption and the journey time (see Figure 11), the traction energy consumption of Strategy 1 is less than that of Strategy 2. Meanwhile, Strategy 1 and Strategy 2 will generate electricity on the LDSD section and feed back to the traction network.In Interval 2, the full electrical braking distance of Strategy 1 is greater that of Strategy 2 and the average electrical braking force of Strategy 1 is also greater than that of Strategy 2. Then we could infer that Strategy 1 generates more energy than that of Strategy 2. Hence, considering the energy consumption of Interval 1~Interval 3, we can conclude that Strategy 1 is more energy-efficient.

Numerical Algorithm
In this paper, the necessary condition of the train energy-efficient operation is derived.The linkage of the speed holding section is given.The constraints ( 13), ( 16) and ( 17) are considered in the optimal control problem.However, to obtain an optimal trajectory of the train operation under the given time, the constraint (15) should be taken into consideration.An iterative algorithm is proposed to calculate the optimal trajectory of the freight train energy-efficient operation incorporating the LDSD section.It's difficult to choose an initial holding speed ( c v ) with PP, in the first iteration, suppose the initial holding speed is: The detailed procedure of the algorithm is as follows: (1) Calculate the shortest time ( 1 T ) from 0 to X, obtain the flat-out running trajectory.• If TT  , then go to step (2).
• If TT  , optimization terminates, the flat-out running trajectory is returned.Analysis: in Figure 12, suppose there are two driving strategies on the LDSD section.
The trip from 0 to X is divided into three intervals as follows.v c and v c are the holding speed (PP) of Interval 1 and Interval 3. T s is the journey time of Interval 2. Given the total journey time T set , the journey time of Interval 1 and Interval 3 is T set − T s .
As ϕ(x 1 ) > ϕ(x 2 ), the journey time of Strategy 1 is less than that of Strategy 2 in Interval 2, which means that the average speed of the freight train with Strategy 1 is greater than that with Strategy 2. Obviously, the number of the pneumatic braking instances with Strategy 1 should be less than that with Strategy 2 because the average speed with Strategy 1 is greater.Therefore, the number of freight trains that pass Interval 2 in the fixed time has increased.Meanwhile, if Strategy 1 is applied in Interval 2, the journey time T s ↓ , then the journey time of the rest intervals (T set − T s ) ↑ .And the holding speed drops down from v c to v c .According to the relationship between the energy consumption and the journey time (see Figure 11), the traction energy consumption of Strategy 1 is less than that of Strategy 2. Meanwhile, Strategy 1 and Strategy 2 will generate electricity on the LDSD section and feed back to the traction network.In Interval 2, the full electrical braking distance of Strategy 1 is greater that of Strategy 2 and the average electrical braking force of Strategy 1 is also greater than that of Strategy 2. Then we could infer that Strategy 1 generates more energy than that of Strategy 2. Hence, considering the energy consumption of Interval 1~Interval 3, we can conclude that Strategy 1 is more energy-efficient.

Numerical Algorithm
In this paper, the necessary condition of the train energy-efficient operation is derived.The linkage of the speed holding section is given.The constraints ( 13), ( 16) and (17) are considered in the optimal control problem.However, to obtain an optimal trajectory of the train operation under the given time, the constraint (15) should be taken into consideration.An iterative algorithm is proposed to calculate the optimal trajectory of the freight train energy-efficient operation incorporating the LDSD section.It's difficult to choose an initial holding speed (v c ) with PP, in the first iteration, suppose the initial holding speed is: The detailed procedure of the algorithm is as follows: (1) Calculate the shortest time (T 1 ) from 0 to X, obtain the flat-out running trajectory.

•
If T set ≥ T 1 , then go to step (2).

Train Operation Simulation under Different Journey Time
The optimization algorithm is implemented based on Section 6.The optimal trajectories under the four groups of the journey time are calculated, which are shown in Figure 14.

Train Operation Simulation under Different Journey Time
The optimization algorithm is implemented based on Section 6.The optimal trajectories under the four groups of the journey time are calculated, which are shown in Figure 14.The track gradient is transformed to the track height for observation convenience.Assume that the track height is 0 km at the origin location, then the track height of other location is equal to the sum of multiplying the gradient value by the gradient length from 0 km to current train location.Now we obtain four optimal trajectories by using the multi-particle model of the freight train.At the start phase, the freight train is running with FP.The velocity of the freight train must be kept less than the speed limit (2 km/h below the speed limit).To prevent the rear of the freight train from exceeding the speed limit, the velocity is accelerated until the last wagon passes the low speed limit (see Figure 14).To make the train operation energy-efficient, the optimal control should be switched from PP to C before entering the LDSD section.In the LDSD section, periodic braking is applied and the optimization trajectories are the same under the four groups of given because the jump condition (see Sections 4.2 and 4.3) should be satisfied.The simulation results of the optimization algorithm are listed in Table 3.The operation energy of different times is listed in Table 3.The total energy consumption is defined as: where J is the total energy consumption, J t is the energy consumption of the freight train operation with FP and PP, J d is the regenerative braking energy of the train operation with electrical braking.In Table 3, the total energy consumption is less than 0, which implies that the train generates more energy than it consumes in the given track interval.Meanwhile, reducing the journey time will cause the energy consumption and total energy consumption to increase.The relationship of journey time and energy consumption/total energy consumption is presented in Figure 15.
The track gradient is transformed to the track height for observation convenience.Assume that the track height is 0 km at the origin location, then the track height of other location is equal to the sum of multiplying the gradient value by the gradient length from 0 km to current train location.
Now we obtain four optimal trajectories by using the multi-particle model of the freight train.At the start phase, the freight train is running with FP.The velocity of the freight train must be kept less than the speed limit (2 km/h below the speed limit).To prevent the rear of the freight train from exceeding the speed limit, the velocity is accelerated until the last wagon passes the low speed limit (see Figure 14).To make the train operation energy-efficient, the optimal control should be switched from PP to C before entering the LDSD section.In the LDSD section, periodic braking is applied and the optimization trajectories are the same under the four groups of given time because the jump condition (see Sections 4.2 and 4.3) should be satisfied.The simulation results of the optimization algorithm are listed in Table 3.The operation energy of different times is listed in Table 3.The total energy consumption is defined as: where J is the total energy consumption, t J is the energy consumption of the freight train operation with FP and PP, d J is the regenerative braking energy of the train operation with electrical braking.In Table 3, the total energy consumption is less than 0, which implies that the train generates more energy than it consumes in the given track interval.Meanwhile, reducing the journey time will cause the energy consumption and total energy consumption to increase.The relationship of journey time and energy consumption/total energy consumption is presented in Figure 15.In Figure 15a, the traction energy consumption drops down while the trip time of the train increases.Less energy is consumed if the train runs more slowly.It seems the train driver should drive very slowly.However, from the railway system's perspective, reducing the journey time not only allows more freight trains to operate on the railway line, but also creates more economic benefit.In Figure 15a, the traction energy consumption drops down while the trip time of the train increases.Less energy is consumed if the train runs more slowly.It seems the train driver should drive very slowly.However, from the railway system's perspective, reducing the journey time not only allows more freight trains to operate on the railway line, but also creates more economic benefit.Dispatchers of the railway should make a compromise between the energy consumption and the trip time.Similarly, the total energy consumption is also monotonically decreasing while the journey time of the train increases even if the regenerative energy consumption is considered.It means that more energy will be fed back to the traction network if the trip time is longer.

Comparison between The Proposed Algorithm and Other Methodologies
In Section 7.1, the optimal trajectories under different journey time are calculated.The simulation result shows that the energy consumption is monotonically decreasing while the holding speed v c declines.Based on the simulation results of the proposed optimization algorithm, the energy consumption converges to a minimum value if the holding speed is very close to 0.
In this section, (FPC) and field operation data are compared with the proposed algorithm under the same trip time (95 min), which are illustrated in two case studies.Figure 16 is the simulation result of the proposed algorithm.Dispatchers of the railway should make a compromise between the energy consumption and the trip time.
Similarly, the total energy consumption is also monotonically decreasing while the journey time of the train increases even if the regenerative energy consumption is considered.It means that more energy will be fed back to the traction network if the trip time is longer.

Comparison between The Proposed Algorithm and Other Methodologies
In Section 7.1, the optimal trajectories under different journey time are calculated.The simulation result shows that the energy consumption is monotonically decreasing while the holding speed c v declines.Based on the simulation results of the proposed optimization algorithm, the energy consumption converges to a minimum value if the holding speed is very close to 0. In this section, (FPC) and field operation data are compared with the proposed algorithm under the same trip time (95 min), which are illustrated in two case studies.Figure 16 is the simulation result of the proposed algorithm.In Figure 16,  3.At the origin location, the initial adjoint variable ( ) is large and it's decreasing quickly with the location of the train increasing.Here,  ranges from −1 to 2. The optimal controls are decided by the value of the adjoint variable.For an example, if  declines from 1 to 1 dt     , then the optimal control switches from PP to C. Applying coasting before entering the LDSD section makes the freight train operation more energy-efficient because more potential energy is converted to kinetic energy.While the train is running on the LDSD section, FEB and FB are applied alternatively because of the jump of the adjoint variable.The force that varies with location of the train is also shown in Figure 16.

Fuzzy Predictive Control
The (FPC) algorithm was designed in [34].There are two controls for the algorithm: the deviation of the current train speed and target speed, the acceleration of the freight train.They are put into the In Figure 16, v c = 25.5 km/h , the operation time is 5704.20 s (≈ 95 min), more details are shown in Table 3.At the origin location, the initial adjoint variable (θ) is large and it's decreasing quickly with the location of the train increasing.Here, θ ranges from −1 to 2. The optimal controls are decided by the value of the adjoint variable.For an example, if θ declines from 1 to αη d η t < θ < 1, then the optimal control switches from PP to C. Applying coasting before entering the LDSD section makes the freight train operation more energy-efficient because more potential energy is converted to kinetic energy.While the train is running on the LDSD section, FEB and FB are applied alternatively because of the jump of the adjoint variable.The force that varies with location of the train is also shown in Figure 16.

Fuzzy Predictive Control
The (FPC) algorithm was designed in [34].There are two controls for the algorithm: the deviation of the current train speed and target speed, the acceleration of the freight train.They are put into the fuzzy logic controller for fuzzy reasoning and optimizing.And the algorithm calculates the control sequences in the future 2-3 km.The simulation result under the given trip time (95 min) is presented in Figure 17.In Figure 17, P = Power.The freight train tracks the target speed before entering the LDSD section.The speed rises up while the train is running on the LDSD section.Periodic braking should be applied to prevent the train from exceeding the speed limits.The train energy consumption for traction is 1821.50kWh.The regenerative braking energy 5492.60 kWh.

Field Operation Data
We obtained the field operation data from Yanqing to Chawu, which consists of the train speed and location.The journey time is 95 min.The speed profile is shown in Figure 18 with P = Power, EB = Electrical Braking, PB = Pneumatic Braking.In Figure 17, P = Power.The freight train tracks the target speed before entering the LDSD section.The speed rises up while the train is running on the LDSD section.Periodic braking should be applied to prevent the train from exceeding the speed limits.The train energy consumption for traction is 1821.50kWh.The regenerative braking energy 5492.60 kWh.

Field Operation Data
We obtained the field operation data from Yanqing to Chawu, which consists of the train speed and location.The journey time is 95 min.The speed profile is shown in Figure 18   In Figure 17, P = Power.The freight train tracks the target speed before entering the LDSD section.The speed rises up while the train is running on the LDSD section.Periodic braking should be applied to prevent the train from exceeding the speed limits.The train energy consumption for traction is 1821.50kWh.The regenerative braking energy 5492.60 kWh.

Field Operation Data
We obtained the field operation data from Yanqing to Chawu, which consists of the train speed and location.The journey time is 95 min.The speed profile is shown in Figure 18 with P = Power, EB = Electrical Braking, PB = Pneumatic Braking.Besides that, the simulation results of the three strategies (optimal algorithm, FPC and field operation) and the energy-saving ratio comparison of the three strategies are listed in Tables 4 and 5.The driving strategy is usually determined by the current train running status and the track gradient, curve, speed limit in the future 2-3 km while a driver is operating the freight train.Meanwhile, excellent driving experience is also significant, which ensures the energy-efficient operation of the train.For an example, an experienced driver will apply coasting before the train enters the LDSD section (see Figure 18).However, the driver operates the train intuitively without calculating the precise traction/braking force and the switching location of the optimal controls.Then 18.6% and 11.4% reduction of the train energy consumption can be achieved respectively compared with that of the field operation and FPC.FPC is a local optimization algorithm which predicts the optimal controls in the finite distance ahead of the train.However, the proposed algorithm in this paper is a global optimization algorithm.We can conclude that the optimal algorithm saves more energy than that of FPC (6.5%).

Conclusions
In this paper, an analytical solution and numerical algorithm are proposed to solve the energy-efficient operation problem considering the LDSD section issue.A new path constraint-auxiliary reservoir air-filled time constraint is taken into consideration in the optimal control model of the train energy-efficient operation.The adjoint variable is utilized to illustrate the optimal control set and calculate the optimization trajectory of the two adjacent speed holding sections.Analysis of the adjoint variable shows that periodic braking (FEB+FB+FEB) should be applied on the LDSD section, which can effectively reduce the operation time on the LDSD section and save the energy consumption of the entire trip.The numerical algorithm is adopted to calculate the optimization trajectory under the given trip time.Case studies indicate that the proposed algorithm in this paper saves 6.5% and 18.6% of the energy consumption comparing with that of FPC and field operation, which has a better performance with respect to energy saving.
The numerical algorithm of the train energy-efficient operation has been implemented in MATLAB.From the perspective of engineering application, the proposed algorithm in this paper could be utilized to calculate the off-line optimal speed trajectory for the driver advisory system.Based on the improved numerical algorithm considering the signal system, temporary speed limit and current train status, the driver advisory system calculates the dynamic optimal speed trajectory and provides the driver dynamic advice about the energy-efficient driving strategies.

Figure 1 .
Figure 1.An analytical schematic of a locomotive's three forces.

Figure 2 .
Figure 2. Traction/Braking force and adhesion force of a locomotive (HXD2): (a) traction force and adhesion force; (b) braking force and adhesion force.

Figure 2 .
Figure 2. Traction/Braking force and adhesion force of a locomotive (HXD2): (a) traction force and adhesion force; (b) braking force and adhesion force.

Figure 2 .
Figure 2. Traction/Braking force and adhesion force of a locomotive (HXD2): (a) traction force and adhesion force; (b) braking force and adhesion force.

Figure 3 .
Figure 3. Mechanical braking force and adhesion force of a freight train: (a) locomotive; and (b)wagon.

Figure 3 .
Figure 3. Mechanical braking force and adhesion force of a freight train: (a) locomotive; and (b) wagon.

Figure 4 .
Figure 4. Multi-particle model of a freight train.

Figure 4 .
Figure 4. Multi-particle model of a freight train.

Figure 5 .
Figure 5. Regenerative energy flow in the traction power supply system.

Figure 5 .
Figure 5. Regenerative energy flow in the traction power supply system.

Figure 6 .
Figure 6.Full electrical braking analysis of the freight train on the long-distance steep downhill (LDSD) section.

Figure 6 .
Figure 6.Full electrical braking analysis of the freight train on the long-distance steep downhill (LDSD) section.

Figure 7 .
Figure 7. Comparative analysis of the optimal controls: (a) FEB and FP; (b) FEB and PP; (c) FEB and C; (d) FEB and FB.

Figure 7 .
Figure 7. Comparative analysis of the optimal controls: (a) FEB and FP; (b) FEB and PP; (c) FEB and C; (d) FEB and FB.

Figure 8 .
Figure 8. Forward and backward speed profile.
the left-hand side and the right-hand side of the train position

Figure 8 .
Figure 8. Forward and backward speed profile.
x dx  ,  jumps from 2   to 2   .Then FP is applied while the freight train is leaving the LDSD.Finally, the freight train will coast to d v .And the optimal control between PEB section and PP section is C.

Figure 9 .
Figure 9. Linkage of the speed holding section considering the LDSD section (v c > V(x)).

Figure 11 .
Figure 11.Energy consumption and journey time.

Figure 10 .
Figure 10.Linkage of the speed holding section considering the LDSD section (v c < V(x)).

Figure 11 .
Figure 11.Energy consumption and journey time.

Figure 11 .
Figure 11.Energy consumption and journey time.
holding speed (PP) of Interval 1 and Interval 3.

sT
is the journey time of Interval 2. Given the total journey time set T , the journey time of Interval 1 and Interval 3 is set s TT  .As time of Strategy 1 is less than that of Strategy 2 in Interval 2, which means that the average speed of the freight train with Strategy 1 is greater than that with Strategy 2. Obviously, the number of the pneumatic braking instances with Strategy 1 should be less than that with Strategy 2 because the average speed with Strategy 1 is greater.Therefore, the number of freight trains that pass Interval 2 in the fixed time has increased.Meanwhile, if Strategy 1 is applied in Interval 2, the journey time s T  , then the journey time of the rest intervals ()

Figure 12 .
Figure 12.Interval time allocation and energy consumption.

Figure 14 .
Figure 14.Optimal velocity trajectories and track height.Figure 14.Optimal velocity trajectories and track height.

Figure 14 .
Figure 14.Optimal velocity trajectories and track height.Figure 14.Optimal velocity trajectories and track height.

Figure 15 .
Figure 15.Relationship between energy consumption/total energy consumption and journey time: (a) Energy consumption and journey time; (b) total energy consumption and journey time.

Figure 15 .
Figure 15.Relationship between energy consumption/total energy consumption and journey time: (a) Energy consumption and journey time; (b) total energy consumption and journey time.

Figure 16 .
Figure 16.Optimal speed profile of train energy-efficient operation considering LDSD section (trip time: 95 min).
time is 5704.20 s (≈ 95 min), more details are shown in Table

Figure 16 .
Figure 16.Optimal speed profile of train energy-efficient operation considering LDSD section (trip time: 95 min).

Energies 2017 ,
10, 794 23 of 26 fuzzy logic controller for fuzzy reasoning and optimizing.And the algorithm calculates the control sequences in the future 2-3 km.The simulation result under the given trip time (95 min) is presented in Figure 17.

Figure 17 .
Figure 17.Speed profile of a freight train based on FPC (trip time: 95 min).

Figure 18 .
Figure 18.Speed profile of a freight train based on the field operation data (trip time: 95 min).

Figure 17 .
Figure 17.Speed profile of a freight train based on FPC (trip time: 95 min).
with P = Power, EB = Electrical Braking, PB = Pneumatic Braking.Energies 2017, 10, 794 23 of 26 fuzzy logic controller for fuzzy reasoning and optimizing.And the algorithm calculates the control sequences in the future 2-3 km.The simulation result under the given trip time (95 min) is presented in Figure 17.

Figure 17 .
Figure 17.Speed profile of a freight train based on FPC (trip time: 95 min).

Figure 18 .
Figure 18.Speed profile of a freight train based on the field operation data (trip time: 95 min).

Figure 18 .
Figure 18.Speed profile of a freight train based on the field operation data (trip time: 95 min).

Table 1 .
Parameters of the freight train in the simulation.

Table 1 .
Parameters of the freight train in the simulation.

Table 3 .
Simulation results under different journey time.

Table 3 .
Simulation results under different journey time.

Table 4 .
Simulation results of the three strategies.

Table 5 .
Energy-saving ratio comparison of the three strategies.